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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.
—from J.R.R. Tolkien's The Lord of the Rings

The Rings of Power are twenty rings with certain magical properties forged in the Elven smithies of Eregion during the Second Age of Middle-earth. All but three were created at least in part by Sauron, who enhanced their properties for his own purposes. Nine of these rings were given to lords of Men, who eventually became the Nazgûl (Ringwraiths), and seven were given to the Dwarf-lords. The One Ring was created by Sauron alone to dominate and command the other rings.

However, Sauron never touched the Three Elven-rings forged by Celebrimbor, who was on to Sauron by this time and hid them from him. They were given to the Elves: Narya, which was originally given to Gil-galad, then worn by Círdan until he gave it to Gandalf; Vilya, which stayed with Gil-galad until he gave it to Elrond; and Nenya, which went directly to Galadriel.

That makes twenty rings. Mary Sues often create a twenty-first ring in order to seem speshul.

Intriguingly enough, Gandalf mentions the creation of several lesser rings by the Elves in The Fellowship of the Ring, but he makes it clear that these lesser rings were made as experiments to practice the art of ring-forging and possessed little power compared to the twenty true Rings; for this reason, Sauron saw little need to corrupt them. (This does not mean that wearing them is a good idea, as they were still too powerful for most wielders to control.) On a similar note, Saruman referred to himself as "Ring-Maker" on one occasion, implying that he wished to create new rings under his control. As this is never mentioned again, it can be assumed that Saruman was unable to create them (luckily for Middle-earth), though from Gandalf's mention of a ring being on Saruman's finger we know he managed at least some progress.

More about the Rings of Power at The One Wiki To Rule Them All.