The cooling system of modern cars is quite simple. The channel network carries liquid antifreeze/coolant around the hotter parts of the engine. The coolant is pressed around the channel by the water pump. The thermostat prevents the coolant from flowing until the motor is hot enough. The rubber hose transports the coolant from the motor to the radiator, as well as the heater core, which is basically a smaller radiator under the dashboard.
The radiator uses external air and a fan to cool the fluid in the system, while the heater core uses heat from the coolant and the fan to heat the air in the car.
In order to quickly heat up the cold engine, it is equipped with a thermostat. When cold, the thermostat restricts the flow of coolant to prevent it from entering the radiator. Once the engine reaches temperature, the thermostat will open and coolant will flow throughout the system. Thermostats, clutches or electronically controlled cooling fans work together to keep the water at an optimal temperature. This is why the thermometer should remain relatively stationary once your car is warmed up.