Faction Warfare simply refers to the rivalries and infighting that occurs between Aeternalae (who are Ovaforms from the Leena Continuum). It can refer to some of their grand scale battles with each other in a Subrealm, or from fighting and competition Factions have for each other on the First Realm. Though such fighting is on a warlike scale, they have street gangster or mob mentality.
Faction Warfare, despite this, is actually done more for the entertainment of the Originae and their subjects than actual progress against each other. After all, eath Aeternal killed just winds up back in the Third Realm anyway and can return right to the fighting. A humorous parallel can be made between Faction Warfare and player "clan" warfare in online computer games.