As Webster 1913 points out, a thermostat works because of "unequal expansion of different metals, liquids, or gases by heat". Your car's Thermostat is no different. It typically uses wax as the agent to operate the heat motor, or the part of the thermostat which converts heat into work - in this case, kinetic energy.
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Basically, there is a spring holding closed a valve on the thermostat. The flange on the thermostat otherwise closes the opening. When the temperature reaches around 180 degrees fahrenheit, the wax melts; going from a solid to a liquid provides even more motion than simply heating a liquid.
In order to test a car's thermostat, you can slip a feeler gauge or other similar thin, flat piece of metal in between the mounting flange and the valve. Now hang the thermostat in a pot of water with a thermometer to watch the temperature. When the water temperature reaches (or at least closely approaches) the intended opening temperature of the thermostat, it should open and fall off of the feeler gauge. Now gaze in awe as the valve opens quite far; sometimes more than an inch.
Replacing a thermostat is generally as simple as removing the housing from the engine via two or three bolts, pulling out the old, and dropping in the new. You will typically also need a gasket. One generally does not apply RTV or other silicone sealant to the gasket, but this is not true of all cars. Check your service manual.
Incidentally, the kind of thermostat most often found "in the wild" (which is to say, in your home) is two strips of different kinds of metal with different rates of expansion (bimetallic strip). They may be simply straight strips bonded together (as in my aquarium heater) or, more frequently, arranged in a coil, as in most thermostats intended for control of heating and cooling. The strip will be seen to curve or deflect itself slightly until it makes contact. The coil, by contrast, provides rotation, and is usually used to control a mercury switch.
In many applications, the thermistor (coupled with some sort of computer control) is replacing the thermostat, because it is a solid state component which seldom if ever wears out. Any "electronic" or "computer controlled" thermostat works on this principle, and has a microcontroller which monitors the resistance of the thermistor to determine its current temperature. This is even used in automotive applications, for example to switch an electric fan on or off.