Sauron's ring was destroyed by Frodo and he was defeated.
How did he create the ring in the first place if he couldn't survive without it?
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It's really a very simple answer:
Sauron is made up of a life force, which before the ring was made, put him at 100% life force. He then decided "Hey, I can put some of this life force into a ring, and then twist it so that I have awesome super powers." So he did. Before you know it, he's going house on people and teaching them what's what.
Then Isildur grabbed Narsil, the sword of his fallen father, and was like "Not on my watch" and wrecked him, taking the ring for himself. Sauron was basically like "oh crap" because like 99% of his life essence just got taken away from him and he basically only has enough to command his troops and manifest as a fiery eye. Even that, however, is tenuous, for it is only doable because out there, somewhere, the rest of his life essence lives on in the form of this super-powered ring.
So, if that ring is destroyed, he loses the ability to manifest at all. Thus being destroyed.
Sauron created the Ring long before his initial death in Numenor.
When creating the Ring, Sauron put "a great part of his own former power" into the Ring (more reading on Sauron's power and the Ring). Because of this, the destruction of the Ring basically destroyed that part of Sauron's power, as Gandalf explains in FotR:
For he will lose the best part of the strength that was native to him in his beginning, and all that was made or begun with that power will crumble.
That destruction of Sauron's power and works is what renders him "a mere spirit of malice that gnaws itself in the shadows" that "cannot again grow or take shape". He no longer has the power to take a form like he does after his previous 'deathes' in the sinking of Numenor and at the hands of Elendil and Gil-Galad.
To further illustrate the answer, imagine being able to take your brain and put it in a box, while still having it functional. Before doing that, you were doing fine, and had no reason to suddenly keel over and die if the box was destroyed, but after having done that, this box is your vulnerability, and destroying it, with your brain inside, will kill you.
This is analogous to what Sauron had done, taking a large part of his essence and magically transferring it into the ring. He wasn't always bound to the ring, only after he crafted it and empowered it. Only then did it become his weak spot.
As to why he did it, it is never explicitly stated, as far as I know, except to hint that in merging his essence with the magical crafting of the ring, he was even stronger than he was alone. The ring, in effect, acted as a power boost, allowing him to control the other rings of power, at the very least. But it also left him vulnerable, as we could see.
Sauron was obsessed with control. Before the ring was created, he was something of an anti-Gandalf or anti-Saruman. He walked the earth, and influenced people through his ability to persuade. However, what he wanted was absolute control. In order to achieve it, he created the ring with which he could influence other ring-bearers directly. In doing so he channeled most of his power into the ring.
It was a risky move on his part, but hey, that's why villains are interesting. They take creative risks.
Just think of voldemort. He created the horacruxs and part of his soul or life was in it. If his body dies he can regenerate through the objects. And before its created he live just like any other person. Creates it before he dies. So when hes killed and the ring is still alive he can come back in another form.But if the ring is ddestroyed while hes alive he is still alive. But if its destroyed whe. Hes dead he cant regenerate.