Rocky was first created in 1946 because many of Bugs Bunny's past foes (such as Beaky Buzzard, Elmer Fudd and Yosemite Sam) were considered very stupid, and that Friz Freleng needed a more intelligent (and challenging) figure to face the rabbit however, Mugsy was first created in 1954's Bugs and Thugs cartoon.
Rocky is a very short man with a very large hat and is also the ringleader of the duo, being smarter than Mugsy, but also having a short fuse- thus, he is prone to violence when he gets angered. His face is seldom seen, as his hat pretty much covers most of his head and, despite his small size, he is a formidable fighter if he has to be and is proficient with firearms (at least for a cartoon character).
Mugsy is large but stupid and, although loyal to Rocky, is often the target of the former's violent temper- despite his size, Mugsy doesn't seem to be as proficient a fighter as Rocky, though it could be that his loyalty to Rocky prevents him from actually fighting back.
The two live the lifestyles of stereotypical criminals and are wanted men supposedly notorious in their setting- though like most (if not all) Looney Tunes villains, they are hapless against the antics of the hero (often Bugs Bunny or some other "main" toon).