Okay, this turned out to be not so much a little run-down, but there was a lot to cover here.

*ahem*

There are twelve major factions in Sigil (also known as 'Ascendent Factions'):

Athar: Believes that the Gods are not in fact 'divine' but are frauds. They have network of spies within Sigil (and beyond) to gain intel on religions and make sure none grow too powerful.

Bleak Cabal: Believes that life is meaningless and cruel, so they do their best to bring some comfort. They set up in the Hive Ward in Sigil, which is the poorest and most crime-ridden part of the city, and live in the Gatehouse which has a sanctuary of the homeless and fugitives, house of healing, an orphanage and a soup kitchen.

Doomguard: Believe that the point of life is its inevitable entropy and destruction. Some interpret this as trying to hasten destruction, some think you should just natrually let it takes its course, and some try to delay it. They oversee the creation and production of all weapons in Sigil.

Fated: Believes that anyone can stake a claim to the multiverse by simply knowing what they want, figuring out how to get it, and being strong enough to hold onto it. They are the city's tax collecters.

Fraternity of Order: Believes that laws are representations of power. Knowledge is power, and knowledge of law is the ultimate power. By understanding how the law works, one can exploit or even break them in the right circumstances. They oversee the city's courts.

Hands of Havoc:  They believe that the status quo is built on lies and greed, with laws only benefiting those in power and keep the people oppressed, and only through a trial of fire (both in the metaphorical and literal sense) can the world be improved. They don't have a set role in the city, they just stir up trouble from minor acts of vandalism to actual arson.

Harmonium: Believes there are only two states to existence: war and peace. When neighbours share the same views, there's peace, but disagreement breeds conflict and instability. The purpose of the Harmonium is to unify the multiverse into a state of peace... by any means necessary. They are the city's law enforcement.

Heralds of Dust: Believes that everyone is already dead, and that this 'life' is simply a form of afterlife. The goal of their members to achieve the 'True Death' and help others along on the same journey. They run the city's morgue.

Mercykillers: Believe mercy is a weakness, and the only way to secure the perfect multiverse is through purging it of wrong through cold, relentless justice. They run the city's prison.

Mind's Eye: Believe everyone has the potential to become a god. Their base is in the city's foundry and they seek to hone their potential and shape the multiverse through crafting. Their members also travel around a lot as part of their journey of self-discovery and self-betterment.

Society of Sensation: Believe the multiverse as a place of experiences - the more varied, the better. By experiencing all there is, one can learn and understand all of existence. They collect various experiences through sensory stones and bring them back to the civic hall to be stored, and those sensory stones can be accessed by the public as a form of entertainment. They also provide other forms of entertainment and promote artistic expression.

Transcendant Order: Believe that for a person to attain a higher state of being, they must learn how to unite their body and mind so that thought and action become one. Those who act purely by instinct can become in harmony in the universe, and their members engage in rigorous training in an attempt to achieve this harmony. They run the city's gym.

[hr]
There are also a bunch of minor factions, also called Sects. Most of them don't operate within Sigil, but throughout the planes. (The Free League, Incanterium and Will of the One are exceptions to this.) There are... a lot of these, so I'm not going to detail all of them, but here are some that are notable or likely to come up in the plot:

Free League: Individualists who believe the other factions are wrong.

Godslayers: A splinter-group of the Athar who take their parent faction's philosophy to see if they can actually kill a god. A large group of them are actually currently camped in the spot where Tir na Og used to be, and claim they are responsible for making the realm disappear. Opinion is divided on whether they're telling the truth, with a majority thinking they're lying.

Incanterium: Believe magic is the source of all power in the multiverse, and constantly hoard magical artefacts and books.

Plane-tenders: believe that all nature, even those in the more abstract planes outside of the Prime, is valid and in need of nurturing and protecting (except for the Abyss and Limbo). Their members all throughout the Planes, but their main base of operations WAS in Tir na Og. The members who weren't in Tir na Og at the time of its vanishing act are understably disgruntled.

Vile Hunt: Believe that talking animals and beast-like races are unnatural and seek their extermination. 

Wylders: Believe that wilderness is the purest state of reality. They and the Vile Hunt don't get along, and are constantly fighting.

Will of the One: A splinter-group of an old Faction called Sign of the One. The Sign of the One believed the multiverse was dreamt into being by 'the One'. While the Sign of the One was ambiguous about who this 'One' is and encourage all its members to believe themselves as 'the One', the Will of the One believes they have found 'the One'. He's some guy called Terwolfe who is constantly ranting insane things, but some of those things have come true so...