Major form of government is the Azorius Senate, though since they're vying for control with other guilds, it makes things tricky.
The city itself is broken up into districts. Most of the events of the actual sets take place in the tenth, which is on top of a plateau where the major guild headquarters are.
Boros are effectively Roman guards crossed with that loose cannon cop who doesn't play by the rules. They function as both military and police.
Azorius also has Gus says and managed which are more disciplined. They don't have true authority over the boros, and can be thought of as closer to FBI.
Selesnya and Orzhov also do a bit of policing. Selesnya are essentially militia in times of need, while Orzhov are the mafia and prefer to keep order in neighborhoods they control.
Most places in the important distracts can be assumed to be under one guild or another's control, but there's a lot of grey areas where they overlap or there's no real claim.
Crime is mostly the domain of the Gruul (naturalist vandals who tear down buildings), Rakdos (crazy murder carnival), Dimir (spies and info brokers), and Orzhov, again (the Catholic church if it was a front for the mob)
Izzet, Simic, and Golgari are the more neutral guilds, being focused on science and agriguclture, though they do have their leanings.
Typically, you can expect your players to want to be members of a guild, as the more without tend to be screwed a lot. It's a good idea to ensure they're choosing from guilds that don't contradict too much though. A look at the color pie can often identify these though.
An interesting note is that the enemy color guilds often overlap with their shared ally color in terms of who they dislike.
For example, Boros is White/Red, whose shared ally is Green. They heavily dislike Dimir, which is Blue/Black, the enemy colors of green, due to their deceptive and cowardly nature.
Ravnica is a hard setting to get a grasp of. I wouldn't suggest it for your first foray