When looking to increase the power output of an internal combustion engine there are several methods that can be employed. One method that is employed (usually with large gains) is forced induction. Forced induction comes in one of two flavors, both forcing more air into the cylinders.
The principle on both is the same, in that there is a turbine in a chamber with 2 ports. The turbine draws the air from one port in the chamber wall and then back out of the other port. The first port is connected to the air filter and the second port is connected to the admission side of the engine (i.e. the carburetor/throttle body etc).
The design of this turbine can vary from a simple fan design to the complicated G-ladder (pioneered by VW).
The size of the turbine can affect its performance. If you use a small turbine then the total amount of air that can be moved will be relatively small compared to a larger turbine. With a larger turbine a much greater amount of air can be moved however it will suffer turbo lag which is when it takes a small amount of time to achieve maximum air transfer (boost). VW have recently tried to solve the problems by creating a turbine that acts as a small turbine at low RPM`s but then physically changes the impeller blade angle using the centrifugal forces as the RPM`s rise (this is used on the PD TDI engine range).
Now back to the title of this node. The difference between a turbo and a supercharger is the way that the turbine is driven;
On a turbo the turbine is driven by another turbine known as an impeller, the impeller is sat in a separate chamber which has again, 2 ports, the exhaust manifold is connected to one port and then the other port goes to the rest of the exhaust system. The upshot of this is that any waste gasses being expelled from the engine get recycled by driving the impeller, this brings its own problems as it can stall the gasses trying to exit the cylinder.
On a supercharger the turbine is again sat on a shaft however in a supercharger this shaft is driven by a belt so the turbine is directly connected to the crankshaft via a belt (usually toothed). The benefit of a supercharger is that lag is almost completely eliminated. The down side to a supercharger is that there is almost always a whining.
See also:
Ram air
Turbo
Supercharger
Turbo stall
Wastegate
Dump valve