Rings of Power

Three Rings for the Elven-kings under the sky, Seven for the Dwarf-lords in their halls of stone,

Nine for Mortal Men doomed to die, One for the Dark Lord on his dark throne

In the Land of Mordor where the Shadows lie. One Ring to rule them all, One ring to find them, One Ring to bring them all and in the darkness bind them

In the Land of Mordor where the Shadows lie.

The Rings of Power were originally a project of the elven-smiths of Eregion headed by Celebrimbor who was descended from Fëanor.

Their purposes were put to use in the Third Age by the elves who used them to preserve the land and to gain insight into the present and the future. Sauron did have a hand in making the Rings of Power, (although the Dwarves claim that the chief Dwarven ring was given to Durin III by Celebrimbor himself) but he never touched the three elven rings which is why they were not tainted by his power. The making of the rings was such an undertaking that when Sauron planned to make the One Ring he had to place a majority of his power in that ring in order to hold sway over the other rings made by the elves. That is why the rings were not used in the Second Age by the elves since Sauron wore his during that period. Sauron did steal the other rings and presented them to the leaders of the races of humans and dwarves in order to control them. This plan easily worked for controlling men, but the dwarves were too hardy and the rings only made their lust for jewels greater.

The rings are the following:

The Three
The Three Rings of the Elves of Eregion were forged by Celebrimbor alone, and were never touched by Sauron. They were called Narya, the Ring of Fire, worn first by Círdan and then by Gandalf; Nenya, the Ring of Water, worn by Galadriel; and Vilya, the Ring of Air, borne first by Gil-galad and then by Elrond. They remained hidden, and the whereabouts of two were not revealed until the end of the Third Age, after the One Ring was destroyed, and the Dark Lord Sauron was overthrown (Galadriel reveals her possession of one of the rings to Frodo in Fellowship of the Ring).

It could, however, be argued that the possession of the three rings was hinted at during the council of the ring in the Fellowship of the Ring, where Elrond advises that that neither he, nor Lórien, nor the Havens have the power to withstand the might of Mordor.

The Seven
Sauron gave the Seven Rings to the dwarf lords (although according to dwarvish tradition the elves gave one of them to the dwarf Durin III). The dwarves used their Rings to establish their treasure hoards, but Sauron, according to portions of the Silmarillion, was unable to force the Dwarven bearers to submit. Indeed, the rings did not even turn them invisible; they were immune to some of the more detrimental of the rings' effects. It is believed that the dwarves' natural hardiness, and the fact that it was only the more powerful dwarf lords who possessed them, made them resistant to Sauron's control, yet allowed them to accumulate treasure. Of the Seven, at the time of The Lord of the Rings all had either been consumed by dragon fire or acquired by Sauron. (Gandalf tells Frodo in "The Shadow of the Past" chapter that Sauron acquired three of them, which means that the remaining four were consumed by dragons). The final of the Seven was taken from Thráin II, who had been captured, imprisoned, and tormented by Sauron (in the guise of the Necromancer) in 2845 of the Third Age.

The Nine
The nine rings for mortal men were those divided amongst those evil-hearted men doomed to become the Nazgûl, the Ringwraiths. None are mentioned specifically throughout the Lord of The Rings save their leader, the Witch-king of Angmar. His second-in-command is named in the Unfinished Tales as Khamûl, the Black Easterling. The early Middle-earth Role Playing games name the eight other Ringwraiths, Er-Murazor (the Witch-king, of Númenórean race), Dwar, Ji Indur, Akhorahil, Hoarmurath (Númenórean), Adunaphel (female Númenórean), Ren and Uvatha, but none of these names are considered canon, and especially the idea of a female Ringwraith is extremely unlikely within the context of Tolkien's work, given that it is only males who play significant political roles.

The One
The One Ring, secretly forged by Sauron in the heart of Mount Doom, had the power to dominate the other 19 Great Rings. His dominion over the other rings was incomplete, but the force Sauron could bring to bear with the Ring was amazing nonetheless. This was due in part to his placing a large amount of his own power into it at its forging; a necessity that later led to his downfall at Frodo's hands.