Fish must have fins AND scales to be kosher. This does provide an interesting dilemma, because some form of eel which have fins grow scales for some periods during the year, although eel is considered treif just in case. Shellfish, such as crab, clam, oyster, lobster, prawns, turtle, shrimp and the like are all treif, as are shark, whale and porpoise. Fish must not be bottom feeders.
Although fish is considered parev (neither milk nor meat), it should not be cooked with meat, although it may be cooked with milk.