Durin is the name of seven Kings of Dwarves in J. R. R. Tolkien's legendarium. They were held by the Dwarves to be the reincarnations of the first one, Durin the Deathless, resembling him in appearance and said to have preserved memories of their 'earlier lives'.
Tolkien took the name Durin, like most of the dwarf names in his work, from Norse mythology. This he later explained by the translation fiction: since Westron was translated with English, the language of Dale, used by Dwarves for their "outer" names, was translated with Old Norse. One way the original Old Norse term durinn can be translated, appropriately, is "sleepy". 
Discussions of the Durins (specifically in The Peoples of Middle-earth) must be read with care, since their number expanded as Tolkien's story developed. The Durin killed by the Balrog in T.A. 1980, for example, was originally Durin III, but Durin VI in The Lord of the Rings. (And a common typo on the Web lists him as "Durin IV".)
Durin the Deathless
Durin the Deathless was the eldest of the Seven Fathers of the Dwarves. Durin was created by Aulë the Vala during the First Age. He was named "the Deathless" as he lived to a far greater age than any other dwarf, and he was revered by all dwarves as the eldest of their race.
Each of the seven Fathers founded a house of Dwarves. Durin's house was called the Longbeards. His people were known as Durin's folk.
According to the traditions of the Dwarves, he was set to sleep alone beneath Mount Gundabad in the north of the Misty Mountains, which remained a sacred place to them ever after. He awakened some time after the creation of the Elves in Y.T. 1050, and according to an early version of the story travelled great distances to other Dwarf kindreds, where other Dwarves joined him. He arrived at the Mirrormere, a lake in a valley below Caradhras in the Misty Mountains, and there founded what became the greatest and richest of the Mansions of the Dwarves: Khazad-dûm (Dwarrowdelf), afterward called Moria (meaning 'Dark Chasm' in Sindarin).
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