The cooling system in most cars consists of the radiator and water pump. Water circulates through passages around the cylinders and then travels through the radiator to cool it off. In a few cars (most notably Volkswagen Beetles), as well as most motorcycles and lawn mowers, the engine is air-cooled instead (You can tell an air-cooled engine by the fins adorning the outside of each cylinder to help dissipate heat.). Air-cooling makes the engine lighter but hotter, generally decreasing engine life and overall performance.
Diagram of a cooling system showing how all the plumbing is connected
See How Car Cooling Systems Work for details.