A corpse is the remains of a monster or human that has been killed, either during the game itself or prior.
Most monsters leave corpses. Exceptions, which just blow up and leave nothing behind, are the lost soul and pain elemental (and, except for his feet, the cyberdemon). The cacodemon's corpse is probably the biggest and most artistically colorful.
Because they use sprites just like most objects, all corpses of the same type of creature look the same, from any direction.
Space Marine corpses may be seen every so often in random places throughout Doom and Doom II, and represent Doomguy's slain buddies. In Doom multiplayer games, they appear when another player is killed.
Gibs are a special kind of corpse that appears when the creature takes extreme damage. As a rule, only the weakest enemies can leave gibs.