Reindeer (known as Caribou when wild in North America) are a type of deer found in the arctic and sub-arctic. Domesticated reindeer are mainly found in Russia and Scandinavia, whereas wild reindeer are mostly found in Norway, North America, Greenland, Iceland and South Georgia.
Both sexes have antlers which (in the Scandinavian variety) for old males fall off in December, for young males in the early spring, and for females, summer. Reindeer eat lichens, mushrooms, tree leaves, sedges and grasses and on occasion have been known to feed on lemmings, fish and birds eggs. During the summer, when their natural climate thaws and the ground is soft and muddy, their hooves become a bit spongy, which helps them to maintain traction in the sticky, sludgy environment. In the winter, the spongy foot pads of the caribou will shrink to expose the rim of the hoof, so that it can cut into the ice and snow and help hold the caribou’s footing.
Santa Claus' sleigh is pulled by flying reindeer. These were first named in the 1823 poem A Visit from St. Nicholas, (which I know by the alternative name 'Twas the night before Christmas') where they are called Dasher, Dancer, Prancer, Vixen, Comet, Cupid, Dunder and Blixem. Dunder was later changed to Donder and — in other works — Donner, and Blixem was later changed to Blitzen. Some consider Rudolph as part of the group as well, though he was not part of the original named work referenced previously, and did not join the sleigh team until later.