If you read the books, and especially the Sillmarillion, you'll find the answer. The fine details are slightly debated (a lot of Tolkien's later writings weren't published until after his death), but here's a summary that will help things make more sense.
Basically, it takes power for the beings that are made of power (like the "Valar" (creators of the world), or the "Maiar" (servants of the world's creators), like Gandalf, Sauruman, Sauron, and the Balrog) to manifest themselves physically, work "magic", or to generally affect the physical world.
About 3,000 years before the start of the Lord of the Rings, Sauron forged most of his power into the One Ring; it magnified his power, and nearly won him the war for Middle Earth. The Elves and Men and Dwarves all banded together, and defeated Sauron, and he was almost completely destroyed; his power was scattered and weak. But the One Ring held his power intact, and so long as it existed, Sauron could not be completely destroyed.
Over the long centuries, he managed to grow in power again (in your words, he slowly "put himself back together again"), and once again became a threat to Middle Earth, even without the power of the Ring. Once he got his Ring back, he would regain all his former power; and take over the world. His victory was all but assured; his greatest fear was that one of the heros of the Middle Earth might try to use the power of the One Ring against him; though in the process it would corrupt the hero, he might succed in first unseating Sauron from his Dark Throne.
For that reason, Aragorn, the true King of Gondor, gathers an army of all that's left of Gondor, Rohan, the Elves, and the Dwarves, and leads them all on a suicide mission to storm the very gates of Mordor; just so that Sauron will concentrate on killing Aragorn and his army, not on on Frodo and Sam, as they silently tiptoe their way into Mordor.
Somehow, despite all odds, the hobbits managed to sneak into Mordor, climb Mount Doom, and destroy the One Ring under Sauron's very nose.
Since the One Ring was sustaining Sauron's existance, when it was destroyed, so was almost all his power. He faded away into a dark shadow, too weak to ever reform again. At the same time, in a nice dramatic climax, the magic spells that kept his armies intact fell apart, the walls of Mordor came crashing down, and everything created or sustained by the Rings of Power fell apart (including Lothlorian, where Galadriel lived, parts of Rivendell, and sections of the Havens ).
The few elves remaining in Middle Earth, having lost the magical paradises they helped build with the Three Rings, decide to finally leave Middle Earth, and go back and rejoin the Valar in the far West. Frodo goes with them; Sam stays to raise his family in the Shire.
Something else that's interesting: while the hobbits seem to get "lucky breaks" far too often, this may be part of their nature, and indeed, part of the reason why they were chosen. Gandalf hints that there are "other forces" at work, and that Frodo was "meant to find the Ring, and not by Sauron"; and subtle magic is very common in Middle Earth. For example, we learn in other books that Gandalf is an immortal just like Sauron, and his magics were based on understanding and knowledge; like knowledge of the future, even when it seems against all odds. As for the vast co-incidence of Gollum "falling" into Mount Doom, it's interesting to note that Frodo, influenced by the Ring into an unnaturally stern mood, once tells Gollum "may you be cast into the fire if you touch \ the Ring\ again". This may or may not be a death curse (it's debated); but if it was, it certainly worked. And yes, I *did* spend too many years reading rec.arts.books.tolkien as an undergrad! Why do you ask! :-)
In reply to Re^2: At the end of the quest, I would hate to face:
by Anonymous Monk
in thread At the end of the quest, I would hate to face: by jacques