There were eight original reindeers. The names are as follows: Dasher, Dancer, Prancer, Vixen, Comet, Cupid, Donner and Biltzen. The last but most famous reindeer is of course Rudolph the Red-Nosed Reindeer.
All of the reindeer are mentioned in the story of Rudolph the Red Nosed Reindeer, and they are as follows: Rudolph, Comet, Blitzen, Dasher, Donner, Cupid, Dancer, Prancer and Vixen.