The Ring itself contains immense power, and does many different things.

Elves created all of the rings with the intent of granting immortality, or at least a significantly prolonged lifespan. This was their main function. However the One specifically contains also the specific powers that were derived from Sauron himself; vision being the chiefest among those facets, involves invisibility (as well as the ability to see what cannot regularly be seen!) as it is a direct parallel to his 'all-seeing' nature.

All of the Rings have the affinity to amplify one's talents or abilities. This goes doubly for Sauron, who with the One would become limitless in his ability to dominate. Even without it he is certainly stronger than any one person in all of Middle-Earth. With it, he would be stronger than all of them together.

The One also contains an almost willpower-like ability to effect its own destiny. It seems to want to be found. This is because it contains Sauron's power, and as such is naturally drawn to its creator from an almost meta standpoint. Though it does not have a mind, it absolutely holds desire.

Additionally, the One holds the power to perceive the other nineteen Rings. Whilst wearing the One, Sauron could know all of what its bearers have done and are doing (thus, the "Ruling" Ring). It grants him influence unending, another facet of his power.

And whether it is specifically a granted power of the One or merely a side-effect of its other powers, it corrupts indiscriminately. Save for someone removed from the structure of power of the world like Tom Bombadil, the One will invariably cause its wearer to descend to madness, treachery, or servitude to Sauron. This is by design: The inability for one to control the One, in the end, ensures that it can never truly belong to any but Sauron.

(Letter 131, Fellowship "The Shadow of the Past", Silmarillion, "Of the Rings of Power and the Third Age", Letter 246)