Sauron started out as a Maia, an angelic being similar to, but lower in rank and power, the Valar. All of these beings could assume a body -- human or otherwise -- or walk "unclad" in Tolkien's words. To them, a physical presence in the world was as essential as clothes are to us: A convenience, quite useful at times and necessary for interacting with other beings, but not an essential part of their being.
Essential to Tolkien's philosophy is that doing evil causes a loss of being. Doing evil to fellow creatures harms the evildoer in deep, essential ways. Since choice of embodiment reflects the nature of Valar or Maiar, as they diminish, their options for embodiment also diminish. This happens to all of the great Bad Guys of his world. Even Morgoth, once the greatest of all created beings is reduced at the last to being unable to manifest in the world except as a large, blackened, brooding presence and there's a distinct chance that if that body was killed, he might not be able to manifest again.
...their [the Valar's] shape comes of their knowledge of the visible World, rather than of the World itself; and they need it not, save only as we use raiment, and yet we may be naked and suffer no loss of our being. Therefore the Valar may walk, if they will, unclad, and then even the Eldar cannot clearly perceive them, though they be present. But when they desire to clothe themselves the Valar take upon them forms some as of male and some as of female; for that difference of temper they had even from their beginning, and it is but bodied forth in the choice of each, not made by the choice, even as with us male and female may be shown by the raiment but is not made thereby. But the shapes wherein the Great Ones array themselves are not at all times like to the shapes of the kings and queens of the Children of Iluvatar; for at times they may clothe themselves in their own thought, made visible in forms of majesty and dread.
When Sauron made the One Ring, he poured the larger part of his being into it. This meany that his options for embodiment was also tied up in the Ring. This didn't matter so much in the Second Age, as he was as yet not wholly corrupt and he was able to appear in attractive, beguiling forms and even to fool Celebrimbor and the elves of Eregion.
Late in the Second Age, he corrupted himself further by triggering the Fall of Numenor. His physical body was killed in that, and he was diminished to the point where he could no longer assume a "fair" body, but his lust for domination showed in his body.
But Sauron was not of mortal flesh, and though he was robbed now of that shape in which he had wrought so great an evil, so that he could never again appear fair to the eyes of Men, yet his spirit arose out of the deep and passed as a shadow and a black wind over the sea, and came back to Middle-earth and to Mordor that was his home. There he took up again his great Ring in Barad-dr, and dwelt there, dark and silent, until he wrought himself a new guise, an image of malice and hatred made visible; and the Eye of Sauron the Terrible few could endure.
During the War of the Final Alliance, Sauron was again defeated and his body killed and the Ring taken. Evidently -- though I don't recall Tolkien saying so explicitly -- Isildur's claiming of the Ring diminished Sauron's connection to it (ownership or control seems to matter a lot) and what was left to him was reduced to a spirit unable to take physical form again.
Isildur lost the Ring, but Sauron did not possess it either. Yet the Ring survived and, in time, Sauron was able to regain much of his former power and rebuild the Black Tower and make one more attempt to dominate Middle-Earth.
When the Ring fell into the fire, the larger part of Sauron's being died with it and what was left of his greatly diminished self was a "mere spirit of malice" unable to ever take form again.
If it is destroyed, then he will fall; and his fall will be so low that none can foresee his arising ever again. 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.
So: Sauron's ability to have a human form depended on his being, and making the Ring, which was created to enhance his power, required that a large part of his being be put into the Ring. Without the Ring, Sauron was diminished and his ability to embody himself was also diminished.
Prior to his making the Ring, he had his full being and had no problem embodying himself however he chose.
