Cars run a number of systems each of which has a specific task, all of these processes function with one commonality. They all in one way or another rely on electricity from the battery to keep them going. While the diesel fuel pumps and diesel injector both run on electrical impulses many other systems rely on electricity as well. Belt driven systems like the power steering pump and water pump may seem to have nothing to do with the electric flow but they rely on the movement of the motor which is caused by combustion started by a spark.
The alternator is the belt driven unit in the motor which generates electricity that can be flowed to the battery in order to keep the unit from discharging all of its power. If the alternator fails the car will not instantly die because the battery retains a charge that can maintain the systems for a short time. Do not expect to make it very far on just the battery since every function in the car is going to be draining the battery of its life.
Electricity or the movement of electrons is created by the alternator by way of a rotating coil driven by the belt of the motor. Around the coil is a larger coil that uses a magnetically opposite metal that forces the electrons to move through the coils and then out of the lines from the unit. This flow of electrons will charge the battery and keep the car going.