For Native Americans, werewolves were known as Skinwalkers, a man imbued with the spirit of an animal. Neither a curse or an affliction, these beings were considered almost shamanistic in nature, and were quite revered. Though generally they turned to true form (as the animal, not as a hybrid man/thing), skinwalkers were born this way, and their children were as well. It could be granted by the spirit (generally, fox, cayote, stag, and wolf were the well known ones) for those who were exceptional; a great hunter, a clever man, one with nature, etc. Supposedly, this also alters their way of thinking to that of the animal. They also can change at will, as oppose to the waxing and waning of the moon.
The Fae, the race of faeries of Druidic lore, were Shapechangers, and could alter their appearance at will. I think this was more magical than physical transformation, but many were known to adopt disguises. Robyn Goodfellow-called-Puck, of Shakespere's Midsummer's Night Dream, turned into an animal, but was still his good old mischievious self.
Changelings are a seperate humanoid race that could theoretically look like anyone/anything, within reason. They could look like a person or an animal (but not a plant). This is more common in Germania, and I believe in Scandinavia as well. Today, the concept of the changeling is now the children's myth, the Boogeyman, a creature that haunts a small, dark place that kidnaps children by preying on their fears.
As for actual Lycanthropy, werewolves were more popular in the European continent, though there are lore in other lands for something similar. Their existance and creation are as numerous as the cultures and nations of our world. Stepping over a grave unknowingly in Armenia might change you into a vampire, so who knows what they might think might make a wolfman? I've heard of a man surviving execution (Russia), a spirit of vengence (some of those cute countries that use to be a part of Russia), a survivor of a wolf attack (Germania/Central Europe), and my favorite, an offender of God (Italy). In the Orient, anything considered a shapechanger was a spirt/possessor. Haven't heard much on African lore or South American lore. If I remember correctly, an Aborigine Werewolf was a predator that could stand up on its own two legs; a reverse-werewolf in a sense.
This probably isn't the best answer to your question, but I wanted to hit upon some of the diversity of what we think of as werewolves.