Gibs is short for giblets, or fowl innards. They 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. Adrian Carmack is credited with coining the term, and Doom is one of the first games that gibs appeared in.
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.