Inside a car engine you'll find a network of waterways through which engine coolant flows. The coolant flows around what's known as the water jacket, then on through the radiator.
Air passing over the radiator dissipates much of the heat, and the cycle continues. For this process to take place the coolant needs to be propelled around the engine, which is where the water pump comes into play.
Technically calling it a 'water' pump is incorrect, because it pumps coolant/anti-freeze – but it's the name everyone knows them by.
The water pump is intrinsically a very simple centrifugal pump containing an impeller within a metal housing – the pump is driven by a belt either from the crank (or occasionally the cam) and the pump's impeller forces the liquid out of the pump.
The action of it forcing the water out, draws more water in. A mechanical pump as fitted to most engines runs continually, or at least all the time the engine is running.
The downside to this is that because it places extra load on the engine it does sap some power, and therefore affects economy.
To combat this, some vehicles feature electric water pumps that can be controlled by the ECU to only operate when needed, and to alter the volume and speed of the flow to optimise the temperatures.
These are considered to be more efficient, and are becoming increasingly commonplace. They are however less likely to be durable over time, as a traditional water pump is a simple mechanical device, as opposed to a complex electrical one.