Fires & Destruction
When starting a new city, you're given a “honeymoon period” where random disasters are temporarily disabled, allowing you the freedom to do your opening moves in peace. However, it doesn't take long before that ends; after about 400 citizens, you'll be at-risk for fires, health, crime, and garbage problems, and after about 1000 citizens, you'll be at-risk for all disasters.
You need to be prepared. A fire in the wrong part of town at the wrong time can decimate large sections of your city, which can seriously slow down your efforts. It's nearly impossible for a city to become completely destroyed; even if the worst comes to pass, you can issue a bond and lean on neighbors for help. However, there's no reason to waste money letting fires burn for no reason.
Unlike previous iterations of the franchise, it's possible for multiple fires to start in different areas of the city at once. If you only have one firetruck, then at least one fire will go ignored and possibly cause serious damage.
Prevention
Fires can start in any building in the city, but the highest chance of fires start in Industrial zones. Uneducated Sims also have a much greater chance of starting a fire, and if you have an arsonist running around, he might start a fire in any random zone that he wants. Low water supplies make it worse (since buildings can't activate their sprinkler systems).
When plopping a Fire Station, you'll see its radius on the roads. These areas are less likely to burn, but that doesn't mean they're immune. If you place a Fire Marshall module with a Large Fire Station, he'll drive around and prevent any fires from forming in that building for a while, but he can't fireproof your whole city.
A HazMat truck puts out heavy Industrial fires caused by hazardous materials. Without it, Hazmat fires will burn unchecked.
Stages of Fires
No matter how well you prepare, a fire will happen, whether because of a natural disaster or because of your Sims starting them accidentally. When a building ignites, it goes through four stages.
Stage 1
At stage 1, the building will begin to smoke. Any Sims who are inside the building (whether they're working or living there) will try to escape. The more residents or workers in the building, the slower they'll be in getting out. Any Sims who manage to get out will be safe, but they'll stand around and watch the building burn.
As soon as the building smokes, all fire stations in the city will be alerted. The closest one will send a firetruck as soon it can, if it has a firetruck to send. (Remember that by default, normal Fire Stations take 12 in-game minutes to dispatch a firetruck, while Large Fire Stations take only 6 minutes.) If you have no fire stations, or all the ones in your city have already dispatched their trucks, any neighboring cities that are volunteering their firetrucks will send them.
The firetrucks will take the fastest route they can to the fire, but they're subject to Traffic. If the roads are congested, it'll delay the trip, and the stages may advance.
If the building is only smoking, it's fairly safe. The fire won't spread, and if the firetrucks can get there, they can put it out quickly. Even better, the building will be saved, and the Sims who escaped will go back inside to live or work as normal. If your streets are clear and your stations are relatively centralized, this will be the result of most of your fires.
Stage 2
At stage 2, you'll see small flames out of the windows and one coming out of the roof. At this point, the fire can spread, but it's unlikely; if it does, it can only spread to neighboring buildings and cannot cross the street. If you think your firetrucks can't get there, you can Bulldoze the buildings around the burning building to create a fire break.
Starting with stage 2, any Sims who were unable to escape the building will become injured, but will still steadily escape. Injured Sims require ambulances, so your Healthcare buildings need to be ready as well. If the injured Sims don't get medical help quickly enough, they will die; remember that your ambulances are subject to traffic as well, so if your firetrucks are having problems getting to the building already, your ambulances probably won't be much better off.
At any point with stages 2, 3 or 4, if the fire is extinguished, the building is flagged as “Abandoned” and will need to be bulldozed. Any remaining Sims in the building will be injured, but will live.
Stage 3
At stage 3, the image looks similar to stage 2, with fires in all the windows. However, at stage 3, the fire plume on the roof will start to smoke. The fire can spread quickly, but still only affects the immediate buildings. Sims who come out will be more seriously injured. Because multiple Sims will come out, and because your Healthcare facilities have already dealt with the victims in Stage 2, your ambulances may be stretched thin.
If your firetrucks aren't at the scene by stage 3, then you have a city-wide problem. Consider lowering your Traffic density with a Mass Transit Strategy, or build another Fire Station closer to the site to prevent it from happening again.
Stage 4
Stage 4 is the final stage. The fires are still in the windows, but the plumes of smoke are coming out of the windows and roof as well. The fire spreads quickly and can jump over the street, as well as other buildings. Sims are critically injured if they can escape, and they'll need immediate attention. Multiple firetrucks may respond, and will definitely be needed to help contain the fires that start elsewhere. If a fire reaches Stage 4 and is left unchecked, entire sections of the neighborhood can be destroyed before it burns out. Consider plopping a Fire Station immediately, even if you have to issue a Bond, and even if you want to destroy it later.
If the fire is not put out, then the fire overtakes the building and collapses. Any remaining Sims inside instantly die. Remember, only you can prevent SimCity fires.