Especially while talking about FPS, the term "gib" is quite common. While I get it's related to a sudden and gruesome death, I'd like to know its actual meaning and where it comes from.
Gib is short for "giblets" - which are chunks of meat, or more specifically:
This term was quite popular among the early id Software games, Doom and Quake in particular, where a particularly damaging blow to an enemy or another player usually yielded a rain of unspecified meaty hunks, rather than an "intact" monster or player corpse.
Wikipedia has a longer definition along with a list of notable gaming gib implementations:
This article also credits Adrian Carmack, one of the founders of id Software, with coining this term, although this statement is unsourced.
As gibs are an integral part of the shooter experience, there are nearly as many examples and implementations as there are shooter games themselves. A few notable examples besides "classic" Doom and Quake gibs are things like: