Spike's link is the best answer so far.
Turbo is basically a fan that spins using your exhaust. The fan blows air into your intake/engine so now your engine is compressing more air than normal. The bigger volume of air is condensed, making a bigger explosion and more power.
A super charge is basically the same thing. It's run off a belt connected to your engine to run a "fan" that blows air into your intake.
TL
R
More air flow into your engine = more compression = more boom = more power.