Why would Sauron make himself a ring that he invested most of his power in? Wouldn't it be better to keep the power in his physical form and not risk leaving his power in a physical form that could be stolen (which did happen when Isildur took it). Because of its being in the form of a ring, its power could be abused by someone else to attack him.
There are multiple layers to this.
First, it is a typical plot device. The Ring is both a "McGuffin" and a weak spot for a seemingly unstoppable Evil.
It could also be argued that Sauron's power needed a focus for his objective of controlling the other rings, and all of Middle Earth. The Ring was crafted for this purpose. So maybe keeping his power in his physical form was not an option.
Here is Tolkien's view on the manifestation of power (courtesy of The Grey Havens website):
Tolkien's view on the use of power is revealed in another one of his Letters:
"The Ring of Sauron is only one of the various mythical treatments of the placing of one's life, or power, in some external object, which is thus exposed to capture or destruction with disastrous results to oneself. If I were to 'philosophise' this myth, or at least the Ring of Sauron, I should say it was a mythical way of representing the truth that potency (or perhaps rather potentiality) if it is to be exercised, and produce results, has to be externalised and so as it were passes, to a greater or less degree, out of one's direct control."
[The Letters of J.R.R. Tolkien #121)
"For in that time he was not yet evil to behold, and they[Elven-smiths of Eregion] received his aid and grew mighty in craft, whereas he learned all their secrets, and betrayed them, and forged secretly in the Mountain of Fire the One Ring to be their master." The Fellowship of the Ring
As the above poster has said, Tolkien seems to imply that Sauron needed to forge a ring of his own if he was going to master the other Three. To me this makes sense- you can't interact with a medium unless you enter that medium. For example, you can't interact with someone else's Nintendo DS unless you have a Nintendo DS of your own.(I know that seems like an odd example but it was the first one that came to mind.) He needed to have a ring in order to interact with the other rings. And in order to control the other rings, he needed to put a larger portion of his own power into his ring than the Elven-smiths put into their rings while they forged them.
The other answers here do a good job of tackling the specifics of the question, but for a fuller understanding it's necessary to go a little deeper.
One of the main themes of Tolkien's work is the passing of the older, mythological ages and the beginning of more modern, historical times. A major part of this theme is the fading of all that is magical from the world.
Those were the Fading Years, and in them the last flowering of the Elves east of the Sea came to its winter. In that time the Noldor walked still in the Hither Lands, mightiest and fairest of the children of the world, and their tongues were still heard by mortal ears. Many things of beauty and wonder remained on earth in that time, and many things also of evil and dread: Orcs there were and trolls and dragons and fell beasts, and strange creatures old and wise in the woods whose names are forgotten; Dwarves still laboured in the hills and wrought with patient craft works of metal and stone that none now can rival. But the Dominion of Men was preparing and all things were changing...
(Of the Rings of Power and the Third Age)
This fading is of particular concern to the Elves; the words of Legolas in FotR (The Great River) describe it very well:
'Time does not tarry ever,' he said; 'but change and growth is not in all things and places alike. For the Elves the world moves, and it moves both very swift and very slow. Swift, because they themselves change little, and all else fleets by: it is a grief to them. Slow, because they do not count the running years, not for themselves. Yet beneath the Sun all things must wear to an end at last.'
This was something that in particular affected the Exiles; they wanted to have the best of both worlds, maintaining thir kingdoms in Middle-earth, but the desire of the West was always within them.
Moreover they were not at peace in their hearts, since they had refused to return into the West, and they desired both to stay in Middle-earth, which indeed they loved, and yet to enjoy the bliss of those that had departed.
(Rings of Power)
A key power of the 3 Elven Rings is preservation - the areas where these Rings were in action for the longest (Rivendell and Lorien) were like little chunks of the Elder Days that had outlived their time. But they only continued to exist in a fragile balance, maintained by the power of the Rings.
Thus it was that in two domains the bliss and beauty of the Elves remained still undiminished while that Age endured: in Imladris; and in Lothlorien, the hidden land between Celebrant and Anduin, where the trees bore flowers of gold and no Orc or evil thing dared ever come. Yet many voices were heard among the Elves foreboding that, if Sauron should come again, then either he would find the Ruling Ring that was lost, or at the best his enemies would discover it and destroy it; but in either chance the powers of the Three must then fail and all things maintained by them must fade, and so the Elves should pass into the twilight and the Dominion of Men begin. And so indeed it has since befallen: the One and the Seven and the Nine are destroyed; and the Three have passed away, and with them the Third Age is ended, and the Tales of the Eldar in Middle-earth draw to their close.
(Rings of Power)
However, this was also a power that Sauron found desirable; maybe at first for honourable reasons (following his (at least) semi-repentance at the end of the First Age) but definitely later on for domination and control. Sauron was not a destructive Dark Lord: his motive was control, ordering things as he saw fit; in that way he was more of a tyrant than Melkor, who was the destructive one (see the various essays in HoME 10 for more on this).
and of all the Elven-rings Sauron most desired to possess them, for those who had them in their keeping could ward off the decays of time and postpone the weariness of the world
(Rings of Power)
So now we come to the point of making the One Ring, and the main reason why is contained in the Ring verse itself:
One ring to rule them all
That's Sauron's motive all in one line: he wanted to dominate, to control, so the Elven Rings (and all 19 Rings were originally Elven in origin) and all that they did would come under his command.
However, because the Elven Rings (and in particular the 3) were powerful, he needed to make an even more powerful device to control them, and in order to do that he needed to put a lot of his own power into that device - hence the One Ring:
Now the Elves made many rings; but secretly Sauron made One Ring to rule all the others, and their power was bound up with it, to be subject wholly to it and to last only so long as it too should last. And much of the strength and will of Sauron passed into that One Ring; for the power of the Elven-rings was very great, and that which should govern them must be a thing of surpassing potency; and Sauron forged it in the Mountain of Fire in the Land of Shadow. And while he wore the One Ring he could perceive all the things that were done by means of the lesser rings, and he could see and govern the very thoughts of those that wore them.
(Rings of Power)
Yes, it would have been better for him if he had not put his power into the One Ring, but it would not have been in accordance with his motive - he would not have been able to dominate the other Rings, and he really wanted that. So he's caught in a dilemma - either he puts his power into the One and gets to be able to dominate the others, or he doesn't and foregoes the domination he so badly desires. In the end that desire for domination made his choice for him, and was his undoing, and with the destruction of the One the last remnants of the Elder Days went from Middle-earth:
White was that ship and long was it a-building, and long it awaited the end of which Cirdan had spoken. But when all these things were done, and the Heir of Isildur had taken up the lordship of Men, and the dominion of the West had passed to him, then it was made plain that the power of the Three Rings also was ended, and to the Firstborn the world grew old and grey. In that time the last of the Noldor set sail from the Havens and left Middle-earth for ever. And latest of all the Keepers of the Three Rings rode to the Sea, and Master Elrond took there the ship that Cirdan had made ready. In the twilight of autumn it sailed out of Mithlond, until the seas of the Bent World fell away beneath it, and the winds of the round sky troubled it no more, and borne upon the high airs above the mists of the world it passed into the Ancient West, and an end was come for the Eldar of story and of song.
(Rings of Power)
As you'll probably have guessed by now, Of the Rings of Power and the Third Age is an absolutely critical work for understanding this part of Middle-earth's history, and reading it is highly recommended.
Sauron is incredibly arrogant. This is the reason Frodo, Samwise, and Gollum were able to destroy the Ring - because Sauron never imagined that anyone could ever resist the Ring's corruptive effects, or that anyone would willingly choose to destroy it.
Gandalf says as much at the Council of Elrond:
"Into [Sauron's] heart the thought will not enter that any will refuse [the Ring], that having the Ring we may seek to destroy it. If we seek this, we shall put him out of reckoning."
- The Fellowship of the Ring, The Council of Elrond
Sauron's reaction when Frodo puts the Ring on inside of Mount Doom confirms Gandalf's prediction:
And far away, as Frodo put on the Ring and claimed it for his own, even in Sammath Naur the very heart of his realm, the Power in Barad-dûr was shaken, and the Tower trembled from its foundations to its proud and bitter crown. The Dark Lord was suddenly aware of him, and his Eye piercing all shadows looked across the plain to the door that he had made; and the magnitude of his own folly was revealed to him in a blinding flash, and all the devices of his enemies were at last laid bare. Then his wrath blazed in consuming flame, but his fear rose like a vast black smoke to choke him. For he knew his deadly peril and the thread upon which his doom now hung. From all his policies and webs of fear and treachery, from all his stratagems and wars his mind shook free; and throughout his realm a tremor ran, his slaves quailed, and his armies halted, and his captains suddenly steerless, bereft of will, wavered and despaired. For they were forgotten. The whole mind and purpose of the Power that wielded them was now bent with overwhelming force upon the Mountain.
- The Return of the King, Mount Doom
For the same reason - his immense, foolhardy pride and hubris - he never imagined that he would ever be separated from the Ring. It simply didn't occur to him.
Was it foolish of Sauron to put so much of his own power into something that he could lose? Perhaps, depending on your personal outlook. Was it necessary that he do so in order to achieve his goals? Probably. The following quotes should help to explain the issue.
Now Sauron's lust [for power] and pride increased, until he knew no bounds, and he desired to make himself master of all things in Middle-earth... He brooked no freedom nor any rivalry, and he named himself Lord of the Earth.
- The Silmarillion: Of the Rings of Power and the Third Age
And much of the strength and will of Sauron passed into that One Ring; for the power of the Elven-rings was very great, and that which should govern them must be a thing of surpassing potency...
- The Silmarillion: Of the Rings of Power and the Third Age
[Sauron] only needs the One; for he made that Ring himself, it is his, and he let a great part of his own former power pass into it, so that he could rule all the others.
- Gandalf, The Fellowship of the Ring
Sauron didn't make the Ring on a whim; there was a method to his madness. More than anything else, Sauron hated the Elves (and to a lesser extent, the Men of Númenor), and he desperately wanted to bring these enemies to heel. To achieve this goal, he misled the Elves into trusting him, and taught the Elven smiths how to make Rings of Power. He himself played a part in making the Nine Rings intended for Men and the Seven Rings intended for the Dwarves, but the Elves refused to let him interfere with the creation of the Three Rings of the Elves - which, as it was later seen, was a very clever move on the part of the Elves.
In secret, after the other Rings had been made, Sauron went to Mount Doom, and there, with the fires of Orodruin, he made the One Ring - the Ruling Ring. With this Ring, he could now control the Men, Elves, and Dwarves who wore the other Rings - or so he thought. Unfortunately for Sauron, as soon as he put his Ring on, the Elves became aware of it, and immediately realized what he had been planning. The One Ring allowed Sauron to know the thoughts of those who wore the other Rings, and this was part of his plan; what he hadn't anticipated was that the Elves who wore the Three Rings were likewise able to know Sauron's thoughts. Being wiser than Men and Dwarves, and because Sauron had not been present when the Three Rings were made (which apparently limited his ability to control the Three), the Elves who wore the Three Rings immediately took them off.
The Elves were furious at Sauron for tricking them, and thus began the War of the Elves and Sauron. Large swaths of Middle-earth were destroyed in the conflict, and when it ended, Sauron still held the One Ring, although the Elves now knew better than to trust him ever again.
Unfortunately, not everyone is as smart as the Elves, and Men in particular were only too willing to believe Sauron's lies. They, and the Dwarves, accepted Sauron's "gifts", with disastrous results. The Men who received the Nine were very quickly consumed by their Rings; the human tendency of lust for power made Men especially susceptible to corruption by the Rings. In almost no time at all, the human kings who wore the Nine became the Nazgûl - Ringwraiths, the slaves of Sauron.
The Dwarves, on the other hand, were not so vulnerable. Being stout and steadfast, with powerful wills, the Seven Rings simply made the Dwarves who wore them even more greedy than normal. Those who wore the Seven didn't fare very well, but they certainly weren't turned into Ringwraiths. In fact, Sauron was so unhappy with the Dwarves' resistance to the Seven Rings that he tried to get them back, presumably so he could redistribute them to Men.
As we all know, Sauron continued to cause trouble for Middle-earth for several thousand years, but was never quite able to realize his ambitions. He found his way to Númenor, convinced most of its inhabitants to worship Morgoth, and finally, he persuaded the king that, if he led an invasion of Aman (where the gods lived), he could obtain the gift of immortality for all mankind. As you might expect, invading what amounts to heaven is not a very good idea, and the invading army was destroyed by Eru Ilúvatar; Númenor itself was also destroyed, and only the few Númenoreans who had remained faithful to Eru Ilúvatar were spared. Sauron, too, managed to escape the cataclysm, but forever afterwards, he was incapable of taking a form that wasn't hideous and terrible in appearance.
He spent a long time recovering from his ordeal, and when he was strong enough, he made war on the Númenoreans who had survived the downfall of Númenor; these Númenoreans were now living in Gondor and Arnor. Sauron's reemergence surprised his enemies, but they quickly assembled themselves in preparation for war. The Elves, who the Númenoreans were friendly with, joined the cause, and so began the War of the Last Alliance.
Sauron's armies were utterly defeated by the end of the conflict, and Sauron himself was nearly destroyed once and for all. The Elven King Gil-galad and Elendil, the King of Gondor, defeated Sauron in personal combat, but were themselves killed in the process. Elendil's son, Isildur, took the shards of his father's sword and sliced the ring finger from Sauron's hand, and Sauron was thrown out of his physical form, totally defenseless. Elrond and Cirdan urged Isildur to destroy the Ring, but he couldn't bring himself to do so. Within two years, the Ring betrayed him, he was killed in an ambush, and the Ring was lost for hundreds of years.
Gollum found it, or rather, his friend Déagol found it, and Gollum immediately murdered him and stole the Ring. Eventually, the Ring abandoned Gollum, and was found by Bilbo. It passed on Frodo, who eventually played a part in its destruction.
The important thing is that, on one occasion after another, the Ring nearly allowed Sauron to rule the world. This is why he made it in the first place. Sauron with the Ring was far more powerful than Sauron without the Ring, as Tolkien explains in his letters:
The Ring was unbreakable by any smithcraft less than his own. It was indissoluble in any fire, save the undying subterranean fire where it was made – and that was unapproachable, in Mordor. Also so great was the Ring's power of lust, that anyone who used it became mastered by it; it was beyond the strength of any will (even [Sauron's] own) to injure it, cast it away, or neglect it. So he thought. It was in any case on his finger.
A moral of the whole (after the primary symbolism of the Ring, as the will to mere power, seeking to make itself objective by physical force and mechanism, and so also inevitably by lies) is the obvious one that without the high and noble the simple and vulgar is utterly mean; and without the simple and ordinary the noble and heroic is meaningless.
But to [make the Ring, Sauron] had been obliged to let a great part of his own inherent power (a frequent and very significant motive in myth and fairy-story) pass into the One Ring. While he wore it, his power on earth was actually enhanced. But even if he did not wear it, that power existed and was in 'rapport' with himself: he was not 'diminished'. Unless some other seized it and became possessed of it. If that happened, the new possessor could (if sufficiently strong and heroic by nature) challenge Sauron, become master of all that he had learned or done since the making of the One Ring, and so overthrow him and usurp his place. This was the essential weakness he had introduced into his situation in his effort (largely unsuccessful) to enslave the Elves, and in his desire to establish a control over the minds and wills of his servants.
- The Letters of J.R.R. Tolkien, #131
As we can see, the question of whether making the Ring was a good idea isn't as easy to answer as you might expect. From Sauron's perspective (before he lost the damned thing), it was the best idea anyone had ever had. It made him far more powerful, and it allowed him to control anyone who wore one of the Nine, or the Three - or, at least, it would have allowed him to control whoever wore the Three, if the Elves hadn't been smart enough to remove the Three before Sauron could take control of them. The Seven were a total failure - you just can't control Dwarves. The Three were a partial success, or would have been, but Elves are hard to fool. The Nine were an absolute triumph.
It is difficult to say more than that with any certainty. In hindsight, of course, the Ring was Sauron's undoing, and in this light, it was a terrible idea from the start. But we don't live in hindsight. Even a being as powerful as Sauron can't predict the future. With the information available to Sauron at the time, it probably seemed impossible that he would ever lose the Ring, or have to deal with the consequences of being separated from it. He might have realized that it could, theoretically, make him a bit more vulnerable, but even if he did realize this, the risk was wildly outweighed by the rewards - especially because his arrogance led him to believe that, regardless of how dangerous it would be for him to lose the Ring, that would simply never happen.
Once the Ring was made, Sauron never imagined that he might lose it; once lost, he never imagined that anyone who found it could resist its corruptive effects and refrain from using it; once found, he never imagined that anyone would choose to destroy it rather than claiming it as their own.
This might seem to suggest that Sauron was foolish all along, but I don't think that's the whole story - let's break that down piece by piece:
Once the Ring was made, Sauron never imagined that he might lose it: Yes, he eventually lost it, but it took a long time and a lot of effort to make that happen. He held on to the Ring for thousands of years, through at least 3 wars, the destruction of vast areas of Middle-earth, the total ruin of an entire continent (Númenor), and finally, he withstood a siege of several years, then came forth to give personal battle to his enemies. He killed most of the leadership of the opposing forces, including an unknown number of immortal Elves, before Isildur, the son of one of his victims, got lucky and chopped off his finger. Even then, with Sauron more or less dead, for lack of a better word, the Ring continued to do his will. It killed Isildur.
Once lost, Sauron never imagined that anyone found it could resist its corruptive effects and refrain from using it: In this, too, the Ring worked exactly as advertised. Isildur had been the first to hold the Ring after Sauron; he wore it in his unsuccessful attempt to flee the ambush. Gollum found it, and spent the next several centuries using it. Bilbo found it, and frequently used it for several decades. Frodo inherited it, and despite Gandalf's emphatic warnings not to use it, he did so anyway, many times. When Frodo shows it to Tom Bombadil, he immediately puts it on, although it has no effect on him. When Sam briefly carried the Ring, he, too, used it, despite also having heard Gandalf's warnings. As far as we know, literally everyone who has ever held the Ring has also used it at least once.
Once found, Sauron never imagined that anyone would choose to destroy it rather than claiming it as their own: Even here Sauron was right. Of course, the Ring was eventually destroyed, but remember how that came about. Frodo was supposed to toss the Ring into the lava. He didn't. He said "The Ring is MINE!", and put it on his finger. Left to his own devices, he presumably would have walked straight out of the mountain and smack into the arms of the Nazgûl. The only reason this didn't happen is Gollum. He wanted the Ring too much to let Frodo walk away with it, so he bit Frodo's finger off and steals the Ring. Whether you attribute what happened next to Gollum's clumsiness or divine intervention by Eru Ilúvatar, the net result is the same: both Gollum and the Ring wind up in the lava. But the important thing is this: No one, except perhaps Eru, chose to put the Ring in the lava. It got there by accident. No one could bring themselves to discard it, let alone destroy it forever
So you can take whatever answer you choose from this. It is a complex issue, and what you decide to believe is largely dependent upon your point of view.
I stumbled onto the following quote in the Wikipedia article on Fëanor, and it contains an incredibly illuminating parallel:
Fëanor, at the pinnacle of his might, "in the greatest of his achievements, captured the light of the Two Trees to make the three Silmarils, also called the Great Jewels, though they were not mere glittering stones, they were alive, imperishable, and sacred."
Even the Valar, including Aulë, could not copy them. In fact, Fëanor himself could not copy them, as part of his essence went into their making.
So if an elf could create objects that exceeded the capabilities of the Valar by investing some of his essence into the creation, perhaps a Maiar could similarly amplify the power of his creation.
Two similar, but more distant examples I can think of: 1) The Balrogs were only slain by great warriors who sacrificed themselves in the fight. In some respect, they put their essence into the fight. and 2) The Ainur who cared enough about creation to enter it became the Valar. They literally put their essence into their co-creation.
So my answer would be that Sauron, by pouring some of himself into the One Ring, amplified its power to an entirely higher level.