Dolphins were added to Minecraft in the Aquatic Update, and these guys can actually be pretty useful. When fed they will lead you to hidden things like Buried Treasure, shipwrecks, or other nice stuff, so it's worth feeding them if you want to fine some good chests.
Anyway, all dolphins will eat Raw Cod, which is the new name given to all generic uncooked fish in the game. So you don't need to specifically find cod, and most fish in the game will drop Raw Cod upon death. Ironically dolphins will also drop this if you kill them, so you can potentially feed them with their own species.
Feeding them is the same as any other animal, and you just got to put it in your active inventory slot and give it to the dolphin. Once you do that, it will lead you to a secret or a treasure.