As a little-know fact, a turbocharger is actually a type of supercharger. Originally, the turbocharger was called a "turbosupercharger." Obviously, the name was shortened out of convenience.
A turbocharger's purpose is to compress the oxygen entering a car's engine, increasing the amount of oxygen that enters and thereby increasing the power output. Unlike the belt-driven supercharger that is normally thought of when one hears the word "supercharger," the turbocharger is powered by the car's own exhaust gases. In other words, a turbocharger takes a by-product of the engine that would otherwise be useless, and uses it to increase the car's horsepower.
Cars without a turbocharger or supercharger are called normally aspirated. Normally aspirated cars draw air into the engine through an air filter; the air then passes through a meter, which monitors and regulates the amount of air that enters the system. The air is then delivered to the engine's combustion chambers, along with a controlled amount of fuel from the carburetor or fuel injectors.
In a turbocharged engine, however, the air is compressed so that more oxygen will fit in the combustion chamber, dramatically increasing the burning power of the engine. The turbocharger is composed of two main parts: the compressor, which compresses the air in the intake; and the turbine, which draws the exhaust gases and uses them to power the compressor. Another commonly used term in relation to turbochargers is boost, which refers to the amount of pressure the air in the intake is subjected to; in other words, the more compressed the air is, the higher the boost.
Although the increase in power is advantageous to the car -- and likely a source of enjoyment for the driver -- a turbocharger has its drawbacks. First and foremost, a turbocharged engine must have a lower compression ratio than a normally aspirated engine. For this reason, one cannot simply put a turbocharger on an engine that was intended for normal aspiration without seriously undermining the life and performance of the engine. Also, a lower compression ratio means the engine will run less efficiently at low power.
Another major drawback of a turbocharger is the phenomenon known as turbo lag. Because the turbocharger runs on exhaust gases, the turbine requires a build-up of exhaust before it can power the compressor; this means that the engine must pick up speed before the turbocharger can kick in. Additionally, the inlet air grows hotter as it is compressed, reducing its density, and thereby its efficiency in the combustion chamber; a radiator-like device called an intercooler is often used to counter this effect in turbocharged engines.