Most cars on the road today are water cooled. This means the heat from the burning fuel is absorbed by a "water jacket" around the cylinders and the dissipated by the radiator. In order for this to work, a water pump must circulate the water through the engine and radiator.

Water pumps fail in two different ways: Water leakage and bearing failure.

Check bearings like this: Wiggle the fan blade (on rear wheel drive, belt driven water pump applications) There should be very little play in the water pump bearing. On engines with electric fans, wiggle on the drive pulley to the water pump. There shouldn't be play (looseness) in the shaft.

Check leakage like this: Water pumps usually start leaking slowly: occaisional need to add water and spots of coolant on your driveway after a hot day can be from the water pump, especially when you can see no obvious other leak (like a hose or the radiator). When you can get to it to see it, there is always a little hole behind the pulley which drives the pump. This hole is to allow the tiny amount of water which normally comes past a seal (a few drops per year) to get out rather than wash through the water pump bearings. When seals fail, water pours from this hole. Often you'll see stains on the pump housing by this hole on a leaky pump.