Supercharger:
They are belt driven so the power is instant, but this causes some power loss since you are powering the supercharger off the engine, much like your A/C. So you have to give a little power to make a lot. You also will loose any good fuel economy that you once had. No matter what you are boosting when push the gas.
Turbocharger:
Is powerd off the exhuast. The exhust drives a turbine wheel that is attached to a shaft which then drives another turbine that compresses air into the cylinder. This way is far more efficient and there is no power loss. You do experience some lage, but that all depends on the size of the turbo, and there are ways to reduce and even eliminate lag such as N2O. You will also be able to keep a descent gas mileage as long as you don't have a heavy foot.
There are superchagers that work like turbos, but overall I feel that turbos are a better option though being more complicated and costing more. They are no less reliable as some might say.
A good site to find out how things work is:
How Stuff Works.