The cooling system on modern cars is fairly straightforward. A network of passages carry liquid antifreeze/coolant around the hotter parts of the engine. The coolant is forced around the passages by a water pump. A thermostat prevents the coolant from flowing until the motor gets warm enough. Rubber hoses carry the coolant from the motor to the radiator, and also to the heater core, which is basically a smaller radiator under the dashboard.
The radiator uses the outside air and a fan to cool the fluid in the system, while the heater core uses the heat from the coolant and a fan to warm the air inside the car.
For the cold engine to warm up quickly, it is fitted with a thermostat. When cold the thermostat restricts the flow of coolant to prevents it from entering the radiator. Once the engine is up to temperature the thermostat opens and the coolant flows around the whole system. The thermostat, and clutch or electrically controlled cooling fan work in conjunction to keep the water at the optimum temperature. This is why, once your car warms up, the temperature gauge should remain relatively static.