At present, the only one that has lasted very long is the Central Hockey League. (I imagine those guys get pretty burned when we keep using their initials to refer to major junior hockey.)
Then you have small-time circuits like the Ligue nord-américaine de hockey, which is restricted to former Q players, and the SPHL, which operates in towns where even Gary Bettman couldn't put a team. Below that you have fly-by-night leagues like the United Hockey League (folded 2010), but I don't think there are any operating at the moment. None of those leagues pay enough to save a hockey player from asking, 'Do you want fries with that?'
I think a few of the Central League teams are affiliated with AHL teams (and so, indirectly, with NHL organizations), but the leagues below that level are strictly wildcat. So there is really no hockey equivalent of a single-A or rookie league baseball team.
The actual hockey played in these places is nothing special, but I like to check in on them from time to time. Teams and leagues come and go so fast that there's always a good soap opera going on somewhere. If you think the situation in Phoenix is a gong show, or that Daryl Katz is a questionable owner for a hockey team, read up on the
Danbury Trashers. You'll feel good about life for a week, just because you were not involved.