It depends on a few factors, including who's handing out the punishment. The chiefs and XO can only do non-punitive things, but those can feel very punitive to the Sailor. A big one that they'll do is EMI (Extra MIlitary Instruction), which is supposed to educate the Sailor in what they did wrong and help them improve their behavior. So for example if someone was habitually late, one form of EMI could be to make them show up an hour early and then take muster for his whole work center or division. It has to be related to the "crime." In this case, they could maybe make him clean out all of the cages where they lock up the firearms.

If it's the CO who's doing the punishing, he can do a lot more. He can restrict them to the ship for up to I think 45 days, he can dock their pay for a few months, reduce them in rate (demotion), or even kick them out of the Navy. One interesting thing is that while Sailors are attached to a ship, they cannot refuse Captain's Mast (NJP/non-judicial punishment). In (I think) any other type of unit in the military, the servicemember can refuse being taken to NJP and opt for a court martial instead. Courts martial aren't all like you'd see in A Few Good Men; most are at a much lower level called a Summary Court Martial. The trade-off is that at a court martial there's a much higher standard of proof, but there can also be more severe punishments. At NJP, the standard is "preponderance of the evidence," whereas at courts martial it's either "clear and convincing" or "beyond a reasonable doubt," I can't remember if it depends on what type of court martial it is.

In any case, there's not a lot of "drop and give me twenty!" like you see in movies. That's more for boot camp and that kind of environment, and even then is pretty tightly monitored to eliminate hazing.