By the miracle of "cut 'n' paste" here's an interesting study of the nature and habits of the Re'em:
Many monsters were created on the Sixth Day, some destroyed during the Flood, some still with us. The re'em is described as a giant even among these strange animals. At any given time, only two exist, one male and one female, because had more of them existed, the world could not support them. No one is certain what the re'em looks like. The sources describe him as fierce, fast, and indomitable. Scholars argue about the number of his horns, some say he has one, like a unicorn or a rhinoceros. Some say two, and he could be related to the giant aurochs (Bos primigenius), a species of a wild ox that became extinct during the sixteenth century. On the other hand, he may be a purely mythological creature, based on the bas-reliefs of the huge Mesopotamian and Egyptian beasts that were unquestionably familiar to the Jews of the Talmudic era.
The re'ems live at the opposite ends of the earth, one in the east, the other in the west, and for seventy years never see each other -- until the day of their mating. Finally they meet, mate once, and then the female kills the male with one bite.
The female becomes pregnant, and her pregnancy lasts for twelve years. During the last year she cannot walk, only role from side to side, and she survives only because her saliva waters the earth around her sufficiently to produce enough vegetation for her support. Instead of giving birth, her stomach bursts open and she dies instantly. However, twins are born, one male and one female. They get up immediately and wander away, one to the east, one to the west.
During the flood, when Noah collected all the animals into the arc, the re'ems came to join the procession. However, because of their giant size, they could not fit into the arc. Yet Noah saved them. One version claims he tied them behind the arc, and they followed it by running and later by swimming. Another version tells that the flood happened just as the young re'ems were born, so they were small enough to fit in the arc.
King David had an encounter with a re'em. When David was still a simple shepherd, he saw a sleeping re'em and thought it was a mountain. He started climbing it, and the re'em woke up and lifted David on his gigantic horns. David vowed that if God saved his life, he would build Him a temple, a building as high as the re'em himself. God heard him and sent a lion. As the lion is the king of the beasts, the re'em bowed to him by prostrating himself on the ground, and David could descend from the horns. Then God sent a deer, and the lion started chasing her. So David was saved from both the lion and the re'em.
Personally, I find it easier to believe in satyrs and unicorns.