A turbocharger is a device that uses the force of the air leaving your exhaust to spin a fan (turbo). That fan then pushes more air into the intake and provides a more ideal air/fuel mixture for a better combustion, that then provides more power. A downfall to this is that you need to build enough pressure with your exhaust to properly activate the turbo in order for you to increase your power to the engine. This is often referred to as "turbo lag" and can be avoided with a valve system (often called a "turbo lag delete" kit) that always provides a reserve of pressure to be accessible as soon as you start to press on the gas pedal.

A supercharger, however, still provides that "boost" of air/oxygen into the air intake, but it uses a mechanical system to pwer it. Usually it is driven by a belt attached to the engine, kind of like your alternator or cooling fan, but can also be driven by an electrical motor. Since this supercharger will always be turning along with the engine, there will be no delay in added power, or "lag".

In most cases a supercharger would be preffered for performance, but a turbo tends to be easier to install on most engines and can be more versatile.