"The fan belt in your car is the belt that turns the fan. END OF WRITEUP.". On most modern cars, this isn't the case.
On older cars, the fan belt did drive the engine fan, the main fan that blows air over the radiator to help cool the engine down (this is separate to the fan that blows air into the cabin to keep you hot or cold).
However, this means that at low speeds, in start-stop traffic, when the engine cooling needs to be at its best, the fan is operating slowly, and at high speeds (eg on the motorway) when the air rushing through the engine is enough to cool it by itself, the fan is operating unnecessarily. Additionally, it means that as soon as you start the engine, the fan is cooling it, at a time when you want the engine to warm up as quickly as possible!
Most modern engines have electric engine fans that are driven by a simple motor off the battery. Rather than running all of the time, they are turned on and off by a thermostat when the engine is getting too warm. This will almost never happen at high speeds, even on a hot day, but will often happen when in slow traffic on a hot day. You can usually hear it starting up and stopping (and it sometimes will run even once the engine has been turned off until it cools down). Some cars even have two engine fans.
That's not to say that cars don't have a fan belt, just that it's a bad name for it. At the least, the fan belt connects the engine to the alternator, and additional "fan-type belts" connect it to the power steering and air conditioning compressor if they are present.
This all notwithstanding, all of mkb's comments on checking, tightening etc the fan belt apply to these other belts.