"You were born. And so you're free. So happy birthday."
-- Laurie Anderson, "Born, Never Asked" (from Big Science)
"Faction" is a loose term for a group of sentients who share common beliefs, appearance and/or technical or artistic practices. This includes enclaves with a geographical 'homeland' (e.g. Elysium), family clan-groups (Chitin Queens), sub-cultures of enthusiasts (Zips), and art movements - to name a few.
Not all the groups listed here would even consider themselves "factions" as such. Some are merely groups who share a common origin (Terrans, for example) or body style (Plurals), and a few are organizations, such as administrative departments within Upwarp. All groups are included here for the sake of convenience, to avoid having to categorize them any more precisely.
All but the loosest of factions have their own rules, written or unwritten. Most have a common goal of some sort, whether modest or grandiose: a peaceful, orderly life; disrupting other people's peaceful, orderly lives; having a lot of sex; eradicating the Strangevirus; bringing about the end of the universe; and so forth.
There are easily hundreds of these groups, with new factions forming as old ones dissolve. However, in keeping with the generally loose and unstructured nature of life on Puzzlebox, most sentients don't even bother to join one, preferring to do as they please with as few constraints as possible.
Even those who adopt the style of a certain faction may not necessarily be associated with that faction; a sentient with the childlike appearance and neotenic mindstate of the Laughing might in fact be a Surrealist intent on mischief and pranking, a local agent of Holotron Industries, or an Experiencer capturing the experience of childhood for recording.
Some of the established factions are listed below.
For those seeking inspiration in understanding how to think like a member of their chosen faction, there are not only the faction's page and past logfiles to refer to, but a set of Strategies as well.