Factions represents various political organizations, empires, federations, and groups within the galaxy - and across all of the different different universes and timelines brought into Star Trek! For example, the Klingon-Cardassian Alliance exists in uneasy peace with the Federation and the Cardassian Union.
With all of the different factions, it’s no surprise that these factions are frequently in dispute with each other. As you become involved in various disputes within the game, you’ll be called upon to make decisions that favor one or another of these factions. Once you’ve completed a dispute, you can resolve the dispute in favor of one of the factions. Doing so will earn you a greater reputation with that faction, and you may also gain special rewards, such as crew members that are associated with that faction.