A turbocharger is a air-compressor driven by exhaust gas. The basic principal is the harness the energy that is lost through exhaust gas. The exhaust gas is re-routed, it spins the turbine, and is then sent out through the "dump pipe." This design principle has inherent issues with turbo lag, as it needs to "spool up" (ie: get air flowing and spinning the turbine) to make power.
A centrifugal supercharger is almost identical to a turbo, except that it is belt-driven. It makes power on the same principles, but it is driven by a belt which is driven by the engine's crankshaft. Centri's lag, but not like turbo's, as they are constantly making boost (albeit less at lower engine RPM). Most supercharger kits from CAPA are based around Centrifugal Superchargers.
Then you have Roots Superchargers. These are commonly known as "Positive Displacement Blowers." Roots Superchargers work by trapping air from the intake side and forcing it out the bottom of the supercharger through a discharge outlet. They are, basically, air-pumps. They are also driven by the crankshaft via a belt. They constantly make boost, which also earns them the name of "Fixed Displacement Blowers," as they move a fixed amount of air per rotation.
Finally, there's the Twin-Screw Supercharger. These, like roots, are 'air pumps' and appear very similar. They work differently though. Twin screw's have a set of rotors or "screws" which suck air in through the back of the supercharger, compress and force it out through the discharge outlet at the bottom. They create less heat then roots, and are probably one of the most efficient designs out there. They do have a problem though, when you are idle or cruising, they still operate, but send no boost to the engine, so they do have some lag issues. They are also driven by the engine.
This is all just off hand, I may be wrong about some things. However, for more in-depth detail, try this link http://www.superchargersonline.com/content.asp?ID=76
Hope this was helpful.