In Tolkien's mythology, the Three Rings are magical artifacts forged by the Elves of Eregion. After the One Ring, they are the most powerful of the twenty Rings of Power.
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The Three Rings were made by Celebrimbor after Sauron, in the guise of Annatar, had left Eregion. These were free of Sauron's influence, as he did not have a hand in their making. However, they were still forged by Celebrimbor with the arts taught to him by Sauron and thus were still bound to the One Ring. Upon perceiving Sauron's intent, the Elves hid the three from him. They were carried out of Middle-earth at the end of the Third Age, after the destruction of the One Ring.
Narya
The first ring, Narya, was adorned with a red gemstone, probably a ruby. The name is derived from the Quenya nár meaning fire. It was also called Narya the Great, Ring of Fire, Red Ring, and it is also known as The Kindler.
According to Unfinished Tales, at the start of the War of the Elves and Sauron, Celebrimbor gave Narya together with the Ring Vilya to Gil-galad, High King of the Noldor. Gil-galad entrusted Narya to his lieutenant Círdan, Lord of the Havens of Mithlond, who kept it after Gil-galad's death. According to The Lord of the Rings, Gil-galad received only Vilya, while Círdan received Narya from the very beginning along with Galadriel receiving Nenya from the start.
In the Third Age, Círdan, recognizing Gandalf's true nature as one of the Maiar from Valinor, gave him the ring to aid him in his labours. It is described as having the power to inspire others to resist tyranny, domination, and despair (in other words, evoking hope in others around the wielder), as well as giving resistance to the weariness of time.
Nenya
The second ring, Nenya, was made of mithril and adorned with a "white stone", presumably a diamond. The name is derived from the Quenya nén meaning water. It is also called Ring of Adamant, Ring of Water and the White Ring.
The ring was wielded by Galadriel of Lothlórien, and possessed a radiance that matched that of the stars; while Frodo Baggins could see it by virtue of being a Ring-bearer, Samwise Gamgee tells Galadriel he only "saw a star through your fingers". (This appears in many editions as "finger"—which sounds more magical, since it suggests that her finger has somehow become transparent—but The Treason of Isengard, ch. 13, note 34, mentions it as an error.)
Nenya's power gave preservation, protection, and possibly concealment from evil because "there is a secret power here that holds evil from the land". However, the fact that Orcs from Moria entered Lórien after The Fellowship of the Ring and Lórien itself had suffered previous attacks from Sauron's Orcs sent from Dol Guldur suggests the power of the ring did not constitute military prowess. It was said that, protected as it was by Nenya, Lothlórien would not have fallen unless Sauron had personally come to attack it. Galadriel used these powers to create and sustain Lothlórien, but it also increased in her the longing for the Sea and her desire to return to the Undying Lands.
With the ring gone, the magic and beauty of Lórien also faded along with the extraordinary mallorn trees (save the one that Samwise Gamgee grew in Hobbiton) and it was gradually depopulated, until by the time Arwen came there to die in F.A. 121 it was deserted and in ruin.
Vilya
The third ring, Vilya, was made of gold and adorned with a "great blue stone", probably a sapphire. The name is derived from the Quenya vilya meaning air. It is also called, Ring of Air, Ring of Firmament, or Blue Ring.
It is generally considered that Vilya was the mightiest of these three bands (as mentioned in the ending chapter in The Return of the King). The exact power of Vilya is not mentioned. However, it is reasonable to speculate that it also possesses the power to heal and to preserve (it is mentioned in The Silmarillion that Celebrimbor had forged the Three in order to heal and to preserve, rather than to enhance the strengths of each individual bearer as the Seven, Nine, and the lesser rings did). Its power of healing may be particularly strong, as Elrond seems to be the greatest healer in Middle-Earth at the time of the Quest. There is some speculation that the ring controlled minor elements, considering the event where Elrond had summoned a torrent of water as the Nazgûl attempted to capture Frodo and the One Ring.
When Sauron laid waste to Eregion, Vilya was sent to the Elven-king Gil-galad far away in Lindon, where it was later given to Elrond, who bore it through the later years of the Second Age and all of the Third. As Gil-galad was the High King of the Noldor elves at the time of the rings' distribution it was thought that he was best fit to care for the most powerful of the three Elven rings.