The main difference between a 'turbocharger' and a 'supercharger' is its power supply. Something has to supply the power to run the air compressor.
In a supercharger, it use a belt that connects directly to the engine. It gets its power the same way that the water pump or alternator does.
While a turbocharger, gets its power from the exhaust stream. The exhaust runs through a turbine, which in turn spins the compressor
Basically, a turbocharger is more efficient because it is using the "wasted" energy in the exhaust stream for its power source. The bad thing is, a turbocharger causes some amount of back pressure in the exhaust system and tends to provide less boost until the engine is running at higher RPMs. Superchargers are easier to install but the price is more expensive.