As of the That ’70s Expansion expansion, there are 60 factions, which can combine to form 1,770 unique pairs of factions.
Each player shuffles two factions together to form their deck of 40 cards. Every faction typically has 20 cards with some combination of minion cards and action cards. Factions have a typical structure, although there are many exceptions. The typical structure of a faction's cards is the following:
Each faction has a special gameplay mechanic that is common among the faction's minions and actions. For example, Zombies come back from the discard pile, Ninjas specialize in destroying other minions, and Ghosts gain bonuses from having only a few cards in their hand.
Each faction has two associated bases, released in the same set. You might say that each faction has two bases, although the relation is mainly thematic: If a faction is not chosen, its associated bases can still appear, and some bases don't really give an advantage to their associated faction, e.g. Jungle Oasis (the Dinosaurs can easily score it, but only for 2 VPs), Temple of Goju (by themselves, the Ninjas won't be able to get back their highest-power minion), Cave of Shinies (the Tricksters can destroy other players' minions, but they definitely don't want to do it there, and their own minions are usually protected against destruction), The Dread Gazebo (the Ghosts don't get any advantage from it and they even have a play-on-base action),...
Astroknights - Ten unique minions that focus on drawing and playing actions that directly increase minion power.
Changerbots - Mobile faction where minions switch between having higher power and utilizing abilities.
Ignobles - Give control of your minions to others for benefits, often before taking them back.
Star Roamers - Return your minions to hand for benefits, and manipulate how your minions are affected, like moving them instead of returning them or having your power-2 minions be affected instead of others.
Geeks - Use your knowledge of the game to disrupt other players' plans. (Previously found in The Big Geeky Box)
Smash Up All Stars - Draw and retrieve cards from your deck and discard pile to get your best cards out as soon as possible and keep them around for as long as possible. (Previously found in the All Stars Event Kit)