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 Tolkien's invented language of 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 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".)