Ten guilds inhabit Ravnica. Each specialises in two of the five colours of magic, and this directly relates to their general philosophy and role within Ravnica. To give a clearer representation for unfamiliar players, I will also include one of the nine Dungeons and Dragons alignments for each guild because, though the whole system is, ugh... debatably unrealistic, it actually fits really well (Wizards of the Coast own both franchises, so I guess I shouldn't be surprised). For those of you who really want to know, there is one guild for every alignment except true neutral and two for that. The guilds are ordered based on their appearance on the tablet of the guilds (shown below).

The Azorius Senate
Colours: White and blue
D&D alignment: Lawful Good
Quote: "Enter the Senate, the seat of justice and the foundation of Ravnican society."
At the top of the Tablet of the Guilds is the Azorius Senate (it is rumoured they payed a small fortune to ensure this). They are the police force of Ravnica. Often they use completely nonlethal methods when in combat, detaining criminals and labelling enemies as no different. They fight by attaining air superiority with both skynights and sphinxes, and using countermagic to prevent a counteroffensive.
The Orzhov Syndicate
Colours: Black and white
D&D alignment: Lawful evil
Quote: "Enter to find wealth, security and eternal life - for a small price up front."
Clockwise to the Azorius are the Orzhov Syndicate. Though they rarely commit actual crimes (and when they do, they tend to hire someone else to do it for them), they will do a lot to get every last coin out of a situation. Their fighting style involves a form of lifedrain, similar to that wielded by vampires and suitable for their greedy outlook.
House Dimir
Colours: Blue and black
D&D alignment: True neutral
Quote: "Enter and become a mastermind of spies, lies and deception."
Next is House Dimir. A shadowy group of thieves, spies and assassins, the Dimir are practically untouchable. The guildless common folk even believe they are a myth. They do little for themselves, instead generally working for hire. Preferring to avoid combat, when they must fight they use stealth and misdirection before striking at once for a killing blow. Many also know basic countermagic.
The Izzet League
Colours: Red and blue
D&D alignment: Lawful neutral
Quote: "Enter, those with the vision to create and the daring to release their creations."
The Izzet League is a guild of inventors. Followers of both theoretical and applied science and magic, they coninuously experiment with whatever they choose. Their combat style involves using fire and lightning based spells in tandem with blue countermagic. Creatures are particularly rare among the Izzet.
The Rakdos Cult
Colours: Black and red
D&D alignment: Chaotic evil
Quote: "Enter, and indulge your darkest fantasies, for you may never pass this way again."
The Rakdos cult is a depraved, psychotic group of serial killers. Though many are completely insane and nothing but a threat to the people of Ravnica, they are also some of the most capable mercenaries and bounty hunters in existence. They fight will aggression, usually trying to end the battle as quickly as possible, even at their own expense.
The Golgari Swarm
Colours: Green and black
D&D alignment: Neutral evil
Quote: "Enter, those who are starving and sick. You are welcome among the Swarm when the rest of Ravnica rejects you."
The Golgari Swarm follow the philosophy of death being a necessary part of life. Providing shelter for those who may go nowhere else, but just as full of sickness and disease. In combat, Golgari mages take advantage of death, using the corpses of fallen summons to enhance the survivors.
The Gruul Clans
Colours: Red and green
D&D alignment: Chaotic neutral
Quote: "Enter and leave the shackles of society behind."
The Gruul clans are a group of destructive raiders with little central authority. They produce little themselves, instead taking resources from more civilised groups. Like the Rakdos, they are very aggressive and speed-focused in combat, going for a heavy beatdown. Creatures are very common among the Gruul.
The Boros Legion
Colours: White and red
D&D alignment: Chaotic good
Quote: "Enter with merciful hearts. Leave with battle cries."
If the Azorius are the police force of Ravnica, the Boros are its military. Intensely zealous and idealistic, it is a very bad idea to get in the way of a Boros legionnaire. Boros mages use large amounts of creatures, defeating enemies by outnumbering them, with air support from the angels at the peak of the Boros chain of command.
The Selesnya Conclave
Colours: Green and white
D&D alignment: Neutral good
Quote: "Enter and rejoice! The Conclave stands united, open to one and all."
Though it has been described as a "brainwashing nature cult", Selesnya represents the pinnacle of unity and altruism. Following this outlook, many of its spells involve helping and healing others, and focus particularly on protection over attack. Lots of creatures are also used, and these often include centaurs and dryads.
The Simic Combine
Colours: Blue and green
D&D alignment: True neutral
Quote:"Enter and comprehend the perfection of orchestrated life."
The Simic, a secretive group within Ravnica, are obsessed with creating the most perfect form of life. They do this by designing ever newer and newer species, then taking most useful qualities and putting them together. The Simic use large amounts of creatures, and use both the other creatures and spells to improvea handful of them, creating a small but very capable force.