Gibs are the little bits of internal organs, flesh and bone that are left when a player or monster has not only died, but has exploded into body parts.
Gib death occurs when, after a character has been damaged, its health is less than the negative of its spawn health (or original hit points). In other words, the character has suffered, in total, more than twice the damage needed to kill it in the normal fashion. This may also cause the monster's body to be thrown very far from its actual location of death. Telefragging also results in gib death.