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Fomors are defined most commonly as species that are part of the Fomorian Army, engaged in war against the humans. Even if a member of one of those species does not fight against the humans, they are still called Fomor (a notable example being the Ogre, Krunk). Occasionally the word is also used to simply refer to those that are not human, though this seems to be a less accurate description.
The Fomorian army is comprised of many different humanoid species. A host of wildlife seem to fight alongside them, perhaps trained by the Fomors (as the Spiders were) or simply incidentally.
The humans engage in war with the Fomors because of a prophecy that states when all the Fomors are killed, their goddess Morrighan will return and take the humans to Erinn, or paradise. At the start of the game, it is not mentioned why the Fomors are fighting the humans.
Spoiler: (Royal Army: God of The Fomors) A funny thing how the History of the Gods Book said that Fomors are superior and powerful, and humans are weak. But it was the "humans" whom defeated Fomor leader, Shakarr, the two guardians of god of time, and Elchulus, the God of Fomors himself.
Spoiler: (Episode 8) The definition of Fomor is blurred when Ingkells and his army take a potion mixed with a herb created out of Fomorian blood, and are stated to become Fomors. This may be because they were no longer human, or because while they were not members of the Fomorian Army they were fighting against the humans nonetheless.
Spoiler: (Episode 9) Shakarr (who is revealed to be the leader of the Fomors on Episode 2) reveals that like the humans, the Fomors are fighting for prophecy. Their prophecy states that when all the humans are killed, their god Cichol will return and take the Fomors to Erinn.