The Elves and Dwarves, among other creatures featured less heavily in Tolkien's works, were originally (by name) mythological creatures of old Norse sagas.
The names of the Norse Dwarves, or dvergar, are listed in the Völuspá, and Tolkien used these names in The Hobbit: Dwalin, Bombur, Fili, Kili and so on. Gandalf (literally translated as "magic-elf") is also the name of one of the dvergar, though Gandalf himself is modeled after the god Odin.
The name Durin was based from the name off Durinn a dwarf from norse myth. The leader of the dwarves Thorin II Oakenshield was possibly a reference to the norse god Thor and his symbol the Oak tree.
The final battle prophesied at the end of the legendarium, Dagor Dagorath (the Final Battle), bears many resemblances to Ragnarök (literally "fate of the Gods"), the battle which destroys the Norse universe. Both battles result in a near-complete destruction of the world, from which a new world is created, inhabited by those who originally lived.
Tolkien's legendarium centers on the history of Middle-earth, which draws its name from the Old Norse miðgarðr.
In the story of Wayland the Smith found in the Völundarkviða, three swan-maidens fly south over a Mirkwood to the place where they meet Wayland (or Welund) and his brothers.