[Boromir88]

What the Ring did for Sauron all we know is that Tolkien said it 'enhanced' his power:

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.'~Letter to Milton Waldman

Another thing to note about this is that the Ring's power was in 'rapport' or bound to Sauron. Therefor without the Ring Sauron could still rebuild his body if it was destroyed (as he did several times). When he lost the Ring after being overthrown by Gil-galad and Elendil, even without it Sauron was able to reconstruct a new body. The only ways to kill Sauron completely (to the point where he would remain a spirit and not able to reshape) would be to

1) Someone masters the Ring and essentially takes the place of Sauron becoming the new 'Lord of the Rings.' This wouldn't be the best idea as you would simply be replacing Sauron with another one...as Gandalf points out to Frodo and why he can't take the Ring:

'No!' cried Gandlaf, springing to his feet. 'With that power I should have power too great and terrible. And over me the Ring would gain a power still greater and more deadly.' His eyes flashed and his face was lit as by a fire within. 'Do not tempt me! For I do not wish to become like the Dark Lord himself...'~A Shadow of the Past

2) To destroy the Ring and undo everything that is tied to the Ring. When the Ring is destroyed, everything that is bound to it instantaneously dies/crumbles. Sauron is killed, The Ringwraiths perish, and the foundations of Barad-dur crumble.

Now why would Sauron want the Ring back? There are probably several answers but the one that comes to mind first is to prevent #1 from happening. Sauron had no fear the Ring was going to be destroyed (why would he? Tolkien tells us in Letter 246 it was beyond the strength of any will to destroy it). However, Sauron's greatest fear was someone finding the Ring and using it's power against him and using the Ring's power to defeat him:

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 control over the minds and wills of his servants.~Letter to Milton Waldman

Sauron's greatest concern was someone would have the power to use the Ring against him and overthrow him...therefor of course he would want the Ring back in his possession as soon as possible.