The mountain man was a general profession/stereotype of the American Pioneer days. While mountain men are still a familiar trope in American mythology, they are much less well-remembered than the cowboys, the pioneers, and even the early explorers and frontiersmen such as Daniel Boone.
Originally, mountain men were hunters and trappers who traveled into the frontier, particularly the Rocky Mountains, and who made a living, such as it was, by selling furs to the big trapping companies at trapper's rendezvous. Mountain men were rugged, independent, brave, capable, and generally seen as loners and outcasts, living on their own in the wilderness. They were less admirable than cowboys (mountain men might actually try to get along with Indians!), but were still folk heroes and the subject of tall tales and dime novels; Jedediah Smith, James "Grizzly" Adams, John Colter, and Kit Carson were well-known mountain men of their day.
The stereotypical mountain man wore a fur jacket and a 'coon skin cap, had long uncut hair and beard, and carried a rifle. The were large and bear-like, somewhat crude and unlearned, but has lots of real-world know-how. Oh, and they were generally white. Of course, this is a stereotype, and there were exceptions to every part of this description -- except perhaps the 'man' part. I have not been able to find any references to female mountain men.
The heyday of the mountain man was between the decades of 1810 and 1850, as the interior of the continent was opened up and the fur trade boomed. Eventually the fur trade collapsed, and so too the trapping trade that supported most mountain men. Mountain men are well known for exiting the trade and setting up trading posts or helping to found new settlements, although others went a less respectable route and joined the gold rush or married into and lived with Native American tribes. And, as you might expect, a goodly number were killed through accident, disease, freezing, or angering hostile Indians.
Mountain men were well-able to live off the land and could do many odd jobs that were valuable on the frontier (acting as guides, hunters, translators, and short-term labor), so the death of the fur trade was not an immediate end to their lifestyle. But as the interior of the continent became more settled and there was less and less call for fur trappers, fewer men entered the profession until eventually the distinction between a mountain man and a hermit disappeared. By the 1880s the mountain man, as an institution, was basically extinct.