Well on some forums they have local mods(certain sections, like say they'd be in charge of just things under Site General, or Ball Pythons). Then they have Global Mods or Super Mods which are in charge of all the sections. I think Super Mod is just renamed from Global Mod, or it may have a bit more authority over things(it's been awhile since I ran forums).
Basically they enforce the rules. Make sure that people aren't causing trouble, or harassing members stuff like that. They can edit threads, delete posts, merge posts, move posts and ban. Depending on how the admins set everything up they have different ways they can ban like: user, IP and what not. Then they have different ban lengths which they can do all except maybe permanent ban because that would be the decision of the admins. The mods & admins are going to have different versions of cpanel, due to the admins being able to make major changes, and it's a long list of things they could do.
That's basically everything in a nut shell. Though it goes far more in detail then that, because I haven't even mentioned the infraction system, what's logged and so on.
So they enforce the rules, and try to keep everything clean.