It's hard to hammer down specific rules since they vary so much from game to game with the different scenarios and all.
As said above there's the common rules that your gun has to adhere to the field limits. And you ALWAYS call your hits. If you're in doubt whether or not you got hit, you call yourself out anyway. The community is small, and players who repeatedly don't call their hits will get known quickly and not be allowed to games. Eye protection rules are always in effect. Generally, most fields will require either paintball or ballistic goggles. Some private games will allow ballistic glasses, but it's best to used sealed goggles.
Generally when hit, you have to put your kill rag (usually a red bandana you carry) on your head and walk back to your respawn where you can either respawn right away, after a time delay, or some other condition depending on scenario rules. If medics are allowed, then you can be healed, but those are usually governed by game-specific rules too.
Otherwise, it's pretty much dependent on the field or game scenario.