The unicorn (from Latin, meaning "one-horn") was reported in the early Middle Ages of Europe as a shy creature with a body like a horse and a single horn growing from its forehead. That's a pretty good description of the rhinoceros, given that they were being reported only by poorly educated travelers who had ventured into the wilds of the African plains and returned to tell about it. The unicorn may also have been confused with certain African antelopes with straight horns that, seen from a certain angle or as the result of a fight, appear to have only one horn.
The unicorn's horn was reputed to have magical powers, as does rhino horn in some parts of the world even today. The long, straight, curlicue horn we think of as being a unicorn horn was actually the tusk of the narwhal—a type of small whale—caught by fishermen who then passed off the horn as the mystical unicorn appendage. But other horns, including the black horns of the antelope, were often passed off as unicorn horns.
I don't know about you, but the narwhal is a pretty damn cool animal for real, as are the swordfish (and blue marlin) and the sawfish.