A zombie is an undead corpse, typically devoid of a soul and with no free will, originating from Voodoo beliefs. In the Fallout world, the term generally refers to ghouls, who very much resemble corpses, often as a derogatory term supposedly originating from pre-war zombie movies, though it's mostly feral ghouls that show the traits of such zombies from these movies, with their hostility to humans and consumption of human flesh.
The first appearance of the word in Fallout was when it was used by Set's ghoul gangsters to refer to themselves, and was pejoratively used towards Children of the Cathedral cultists. The name is also used in some random encounters with ghouls.
In Fallout 3, most ghouls are very offended by the term. It is commonly used by the bigots who live at Tenpenny Tower, and calling Roy Phillips or Mister Crowley a zombie may lead them to attack the Lone Wanderer. Mister Crowley will hire the Lone Wanderer to kill supposed ghoul bigots by shooting them in the head like they do in old zombie movies.