Reindeer, the usual name in Eurasia, or Caribou, the usual name in North America, are large deer native to Arctic
and tundra/taiga areas. They stand 4-5 feet at the shoulder and have been semi-domesticated by northern-dwelling humans
for hundreds of years for milk/meat/fur and as pack/dray animals. Totally wild herds of both Reindeer and Caribou also still exist.
Unusually for deer, Reindeer/Caribou females grow antlers as well as the males.
The males' antlers are bigger and are sometimes palmated.
In fact, if Rudolph had antlers in mid-winter she was probably female since males lose their antlers in December
and females lose them in the summer; he could have been a young male.
The antlers are re-grown each year, the new growth being covered by "velvet" (above) which carries blood vessels and is shed when the antlers are grown.
Female reindeer with antlers in December
Reindeer moulting its thick winter coat