When Isildur cut the One Ring from Sauron's hand, he returned again. But when Frodo went to Mount Doom and accidentally Gollum cast the Ring along with himself, Sauron was defeated and never came back again. Why is this?
Sauron passed a tremendous amount of his own native power as a Maia into the Ring he forged. Gandalf outlines the situation to Frodo in "The Shadow of the Past":
He 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.
Tolkien, the Old English scholar, uses "great part" here. The Oxford English Dictionary has this gloss (with citations going back to Middle English) for a subsense of great
c. With the. That constitutes more than half, larger, as the great body (of), the great part (of), etc.
indicating specifically more than half. However, as dfri noted in a comment, here Gandalf says "a great part," not "the great part," which only indicates a large portion of Sauron's power, not necessarily the majority. However, he does later clarify that it was indeed the majority. (See below.)
So Sauron placed most of his natural power in his artifact. And actually, it seems likely that he placed almost all of his power in it. The power that went into the Ring was multiplied by the physical instrumentality of the Ring. Once Sauron put it on, he was more powerful than he ever had been before, and to get the greatest advantage from the Ring, it would be natural for him imbue the Ring with as much of his power as possible, so as to get the largest multiplier effect.
The situation changed significantly when the Dark Lord lost possession of the One Ring. Without it in his possession, he could not use its powers to their fullest potential. However, as long as the Ring existed (and no other power had managed to claim and master the Ring), Sauron still benefited from its powers. The foundations of Barad-dûr were secure, and the Ringwraiths were his dutiful servants.
When the Ring went into the Cracks of Doom, all the power that went into it was completely dissipated. Up to that point, Sauron had managed to maintain his position though his distant communion with the Ring that contained so much of his native power. When that power was dispersed, he was not left with enough to have any remaining power over the world. While he was not quite dead, he was reduced to a powerless husk. Gandalf again, in "The Last Debate":
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, and he will be maimed for ever, becoming a mere spirit of malice that gnaws itself in the shadows, but cannot again grow or take shape. And so a great evil of this world will be removed.
Note that here, the wizard says unambiguously that "the best part" of Sauron's power went into the Ring.
Sauron has actually physically "died" at least 3 times:
- Lúthien "killed" him when she overthrew his fortress on the Isle of Werewolves in the First Age.
- At the Downfall of Númenor.
- When Isildur took the Ring from him.
There is a distinction between physical and spiritual "death" in Tolkien. As a Maia, and similar to the Elves, Sauron's spirit is actually bound to the world, cannot leave the world until the End, and can be re-bodied in certain circumstances.
Sauron did not die when the Ring was destroyed
His spirit remained bound to the world, but greatly diminished and no longer capable of much. This is explicitly stated in the text.
It's stated quite clearly that Sauron put the greater part of his own essence into the ring, and although being away from the ring weakened him, it did not do so catastrophically; that part of his essence was still around to keep him strong enough for his purposes.
When the ring was destroyed, that part of his essence was either destroyed (or so it was assumed by everybody who knew about these things) or it was removed from the world entirely so that for all purposes within Arda, it may as well have been destroyed.
It's sort of like putting 90% of your money into the stock of a company. As long as the company is in good shape, for all intents and purposes you are still as rich as you were. But if the company goes bankrupt, such that the stock becomes worthless, then you truly plummet to 10% of your former net worth.
Sauron did not "Die" as a Maiar or spiritual being, he's immortal and cannot die. He invested the vast majority of his essence and identity into the ring he crafted. When it was destroyed the vast majority of his capacity withered with it, thus Sauron no longer had physical dominion over his form any longer.