A haunch is the back end of an animal — its rump and rear leg. When you walk behind a horse, it's important to stay far enough away from its haunches that you won't get kicked.
You can use the word haunch for the hind parts of a four-legged animal or for the hip and thigh of a person. When you squat like a catcher in baseball, you sit on your haunches, and a cow with an itchy backside might rub its haunch against a fence post. The word stems from hanche, which means "hip or thigh" in Old French, from a Germanic root.