A faction is one of 15 philosophical organizations headquartered in Sigil. Each faction believes something different, with the understanding that in the Outer Planes, belief shapes reality. While in Sigil, each faction has taken ownership of a different aspect of the smooth operation of the city's function. The factions engage in a sort of cold war over the beliefs of Cagers, called the kriegstanz.
A "faction" without a headquarters in Sigil is referred to as a Sect.
Outsiders (or "Clueless"), those visiting from the Prime Material plane, are sometimes considered a pseudo-faction (of course, not one with any political sway). Once they've been around and have shaken off their ignorance enough to join a faction, they're no longer considered Clueless.
In modern Sigil, factions seem to be divided into two general classes: Ascendant factions are those larger factions more involved with the operation of the city, while minor factions are those who either more insular or simply don't much care about the politics of it all.
The governing of the city, such as it is, is done at Hall of Speakers. Each faction -- at least each Ascendant faction -- has representation within the Hall, typically that faction's factol. There the factions can discuss, argue, negotiate, and collectively come up with new laws for Sigil. The laws are then enforced by the Harmonium, Fraternity of Order, and (if applicable) the Mercykillers.