basically both are forced induction devices.
That is they blow pressurized air into the combustion chamber and mix it with more fuel. More fuel and air detonating in the cylinder = more power.

A turbocharger is a turbine that runds off the exhaust to produce boost, a supercharger (there are different types) runs off of a belt that does the same thing, makes boost.

That's a very basic explanation, since when do they only apply to imports? :eh: