The long answer:
Car engines are designed so that they operate best when they're very hot. Since they work by having fuel and air explode (literally) in the cylinders, if the engine is cooler, it takes more fuel to get a good explosion. But if the engine is hot, the fuel is easier to explode.
Of course, it is also possible to make the engine TOO hot. This is worse than too cold. If it's too cold, your car's computer adds more fuel, and your car runs. If your engine is too hot, it can warp the metal and cause a lot of damage.
So your car has a radiator with coolant. The coolant is a liquid that boils at a VERY HIGH temperature. It flows through your engine, picking up heat (but not boiling), and into the radiator.
The radiator is metal, which conducts the heat of the collant into the radiator fins. The radiator has lots of fins and, as you drive, air comes in the front of the car and passes over the fins, which cools them. As long as their is a good supply of air flowing over the radiator, your car's engine wont' get too hot.
But sometimes you're stuck in traffic, or driving in a VERY hot area, and more air is needed. So your car has a fan that sucks air across the radiator, helping cool it. But you don't want the car TOO cool, so the fan doesn't run constantly (at lower temperatures, it wouldn't help as much and is a waste of electricity and gas mileage.. Though not a whole lot).
The coolant relay turns the fan on when the car computer senses that the coolant is too hot. The computer sends a signal to the relay, the relay closes, and allows current to flow to the fan, making it run.
Hope that helps