The more powerful the weapon (power being determined in a fairly arbitrary way at times, but that's the intent) the higher the level of licence you need.
I've probably got the two highest level licences a civilian can get without too much difficulty - I hold a Category D licence and a general category handgun licence.
Category D allows me to own most semi-automatic rifles and shotguns. My handgun licence allows me to own a non-automatic handgun that is .38 calibre or less and has a barrel length of over four inches.
(A category C licence allows small magazine semi-automatic weapons, a category A/B allows single shot longarms - hunting weapons like .22s and shotguns for duck hunting. A Category E licence allows for machine guns and similar - technically it's supposed to be possible to get such a licence as a civilian, but I've never heard of anybody doing it. There are also licences to allow people to own handguns outside the 'general category' but again they are unusual).
Are the laws fairly uniform, or do they vary from state to state?