Anything that pushes more air into an engine than it can breath naturally, such as an electric fan, a Compressed air bottle or a Roots type Air mover, is a supercharger. Technically this includes exhaust gas driven turbine Air movers, but these are invariably classed as "Turbochargers" in this day and age.
The act of pushing more air into the engine than it can breath naturally means that you can also get more fuel in as well. More fuel/air means more HORSEPOWER!
Superchargers can be classified as one of two types, Fixed-Displacement and Non Fixed-Displacement.
Fixed-Displacement superchargers, such as the Roots, pump a specific volume of air per revolution. The air comes in through the intake, which is then closed, the outlet is then opened and the air is discharged. At no time throughout the cycle can air pass back through the supercharger.
One way to think of this is by picturing a simple matchbox, now imagine that the inner tray is approx half the length of standard but the same in every other respect, i.e. still a perfect fit when sliding into the box outer. Now imagine lots of the inners being pushed through the box outer, one after another. The air is being transported through the box in each inner, so whilst inside the box it is trapped. This simulates a revolution of a supercharger. The same amount of air going in comes out. The air can not go backwards because of the tight fit of the inner to outer and because the inlet and outlet are never both open at the same time. The box inner needs to move from the inlet side to the outlet side before the air can be discharged.
Non Fixed-Displacement Superchargers
With Non Fixed-Displacement Superchargers (Such as the Centrifugal) an unspecified amount of air is pushed into the engine, much like the effect of a fan blade. The faster the fan blade the more the air is pushed. But unlike the Fixed-Displacement model the air can flow backwards if conditions warrant.
Supercharging is an easy bolt-on way to make big horsepower gains on a relatively standard engine, provided you do your homework and pick the right blower for the job.