What does it mean when the fish in your aquarium lock their mouths?

The fish in your aquarium are cichlids, correct? This means you might have some baby cichlids in your future. It also means you're very good at tank maintenance, because cichlids won't start their spawning rituals in a badly maintained tank. What you are seeing is called jaw locking, and it's part of the cichlid mating ritual. Cichlids don't want to mate with weak partners, so they lock jaws and wrestle. One of two things will happen: one fish will prove too weak to breed, and the victor will kill the loser; or they'll be satisfied. Let's assume the outcome of the jaw-locking contest is good--that both fish are suitable parents. Next you should see the two fish prepare a spawning site. Some cichlids like a flat stone, others the inside of a flowerpot. They'll clean it well, spawn over it, then guard it until the eggs hatch and finally guard the fry. At this point, find out what cichlid you have and what their babies eat. You'll need some of it very soon.

If they are guoramis they are probably fighting.