Sauron poured an element of himself into the Ring. Sauron is a Maia, which is an immortal spirit on the rough level of an archangel in power. He took a part of his inherent self and put it into the Ring of Power (this is why destroying it weakened him from the lord of a vast, dark empire, to a voiceless spirit of malice that could do nothing worse). The other Rings had powers that varied - the Nine Rings given to Men gave them great strength in battle, and wisdom - but they took as much as they gave, and eventually were turned into the Ringwraiths as they drew more and more of themselves from the Rings. The Seven Rings of the Dwarves allowed them to attract wealth, but Dwarves proved strong enough that they did not fall to corruption as Sauron had hoped.
The Three Rings of the Elves were made in secret, away from Sauron, and thus didn't have any chance to corrupt their users. Their powers were in preservation, and healing, and sustaining good things. The three havens of the Elves in the Third Age - the Gray Havens, Lothlorien, and Rivendell - had a Ring of Power at their center (though Cirdan the Ship-wright, of the Gray Havens, gave his ring to Gandalf long before the War of the Ring). Unfortunately, they were made using lore and crafts taught by Sauron (in his magically beautiful form as Anatar, Lord of Gifts). Because of this, the Three Rings were just like all the others - subservient to the One.
When Sauron put on the One Ring, all those using the others fell under his control somehow. The Dwarves lost their rings when Sauron called to them, because they were not ruled by them; the Men fell and became Nazgul; the Elves' minds were totally opened to Sauron, though they did not fall to him. In addition, making the Ring the focus of his power was a way of focusing it all into a sort of lens or the like - he was stronger with the Ring than he was without it, both because of the control of the other Rings and because of the power of the Ring itself.
Complicated topics explained, simply.