Nickname(s): "Defiers", "The Lost"
Headquarters: the Shattered Temple, the former temple of the dead god Aoskar.
Municipal Role: Unofficial Secret Police
Philosophy: The Athar deny not only the gods' right to pass judgment over mortals, but their very divinity. They claim that the gods (whom they call "powers") are powerful but have limits and do not deserve worship. Instead, Athar priests channel divine power from what they call the "Great Unknown", or what they believe to be the true divine force behind everything.
Earth Equivalent: atheists, agnostics, and Deists.
Believers of the Source
Headquarters: the Great Foundry
Municipal Role: (Unofficial) Peace Keepers
Philosophy: This faction believe that each life is a test, and that every person has the potential to become a god.
Earth Equivalent:Hindu and Buddhism (the ultimate goal is not Nirvana but apotheosis).
Nickname(s): "Bleakers", "Madmen"
Headquarters: the Gatehouse (Sigil's insane Asylum)
Municipal Role: Charity and alms
Philosophy: The Bleak Cabal deny that any belief system has any merit; as they see it, the universe has physical rules, but no metaphysical or philosophical ones, therefore any meaning in life must come from within.
Earth Equivalent: Existentialism and Nihilism.
Headquarters: Sigil's Armory
Municipal Role: City Guard, maintaining the Armory.
Philosophy: The Doomguard believe in the sanctity and inevitability of entropy. They see the decay and destruction of the universe as necessary; for once it is destroyed all imperfections will be gone with it, paving the way for a perfect new world.
Earth Equivalent: Cyclic Renewal
Nickname(s): "The Dead"
Headquarters: the Mortuary
Municipal Role: Morticians
Philosophy: Dustmen believe that both life and death are false states of existence, that there is a state of True Death which can only be accomplished by denying one's emotions and physical wants and needs (a conception similar to oblivion, but also conceivably to Nirvana).
Earth Equivalent: Acosmism, Buddhism and Stoicism
Nickname(s): "Takers", "The Heartless"
Headquarters: Hall of Records
Municipal Role: Tax Collectors and record keepers.
Philosophy: The Fated believe that those with power and ability have the right to own what they control and to take what they can from those who are unable to keep it, and that it is their right to exploit any situation to their advantage, regardless of how it affects anyone else.
Earth Equivalent: Social Darwinists and the philosophies of Max Stirner
Fraternity of Order
Headquarters: the City Court
Municipal Role: Judges and advocates
Philosophy: The Fraternity or Order believe that knowledge is power; they learn and exploit both the natural laws of the universe and the laws of society.
Earth Equivalent: Sophists
Headquarters: the Great Bazaar (unofficial)
Municipal Role: None
Philosophy: Those who reject the other factions and their bureaucratic, hierarchical dogmatism; in fact, they don't consider themselves a faction at all. They believe in individual freedom as the highest good.
Earth Equivalent: Libertarianism
Headquarters: City Barracks
Municipal Role: Police Force
Philosophy: The Harmonium believe that peace and stability can only be established under one rule — theirs.
Earth Equivalent: Authoritarianism, particularly religious evangelicalism and fundamentalism.
Nickname(s): "The Red Death"
Headquarters: Sigil's Prison
Municipal Role: Gaolers and executioners.
Philosophy: Mercykllers believe in justice and retribution at the expense of all else. Their name does not come from "killing out of mercy," but rather "killing mercy." Their credo that mercy is for the weak, and the merciful should be punished.
Earth Equivalent: Retributivism
Headquarters: No factol or headquarters. Based around a cell structure they meet at safe houses and secret meeting places.
Municipal Role: (Unofficial) Troublemakers and rabble-rousers.
Philosophy:The Revolutionary League believe that social order and man-made laws are inherently corrupt and must be destroyed—though none of their members can agree on what, if anything, should replace them.
Earth Equivalent: Anarchism
Sign of One
Headquarters: the Hall of Speakers
Municipal Role: Legislators
Philosophy: Adherents of the Sign of one believe that everyone is the center of their own reality and that reality can be reshaped by the power of imagination.
Earth Equivalent: Solipsists
Society of Sensation
Headquarters: the Civic Festhall
Municipal Role: Entertainment
Philosophy: The Society of Sensation believe that accumulating experiential knowledge through the senses is the only way to achieve enlightenment.
Earth Equivalent: Epicurianism and Empiricism.
Headquarters: the Great Gymnasium
Municipal Role: Advisors
Philosophy: The Transcendent Order believe that by tapping in to the 'cadence' of the planes and acting through pure instinct they can achieve a higher state of being.
Earth Equivalent: Taoism and Zen Buddhism.
Headquarters: the Hive
Municipal Role: Chaos and social dynamism.
Philosophy: Xaositechs believe that the only truth is revealed in chaos. The Xaositects have been quite accurately described as being "totally off their rockers, every one of 'em."
Earth Equivalent: Accidentalism, Cynicism and Discordianism.
Most of this information is taken from https://en.wikipedia.org/wiki/Faction_%28Planescape%29
For further information see:
A Player’s Guide to the Planes
The Planescape campaign setting, pages 64-67