A turbo charger uses the presrure from the exhaust gasses to spin a small turbine-like piece. This adds power and allows some of the exhaust gasses to be more thoroughly burned (more power + cleaner emissions).
A super charger forces in outside air which turns some device (the design differs, you may have heard of "roots type", etc...) and effectively does the same job as the turbo, but before-hand, rather than after.
From what I know about this, superchargers can typically be harder on an engine (some, like the motor in the SuperCoupe T-Bird, experience head-gasket problems) due to higher boost pressure than is typical from a turbo.
P.S. From a performance standpoint, a supercharger takes effect basically immediately, while a turbo takes some time to "spool up", producing a delay in boost which is known as "turbo lag" which has, thankfully, been greatly reduced with most modern turbos.
Boost is dependent on the superchargers or turbochargers being used, and problems that may arise from being boosted come in engine design. A supercharger can put out the same psi as a turbocharger.
Most turbochargers are used on either smaller engines (I4's like in Civics) or larger Diesel engines (Turbodiesels are some of the most efficient internal combustion engines available). Superchargers are typically used on V6's and larger V8's. I don't know the exact reason, but I suspect this is because boost comes faster on superchargers, and therefore the power peak happens sooner. The larger the engine, typically, the lower the rev limiter, so it's good to have power low. With I4's and smaller (like those on motorcycles), the rev limits are much much higher, and although the "boost lag" on turbochargers is higher, on an engine that redlines at 7grand or higher, this means less because the power peak is probably near the redline anyway.
Also, but this is just my imagination, I suspect that on smaller engines like I4's, the parasitic loss due to supercharges would matter more than on a bigger engine. And, once again, the different power peaks.
Of course, this is not to say that all turbos have lag; if you get a very small spool, the rotational intertia is lower and the turbo spins up faster. Of course, this means less up top power, but some people have fun and put two turbo's in there; one with a smaller spool and one with a larger, one for low end power (quick) and one for high end power (long).
Same with me, point out what I said wrong if I said something wrong. Howstuffworks.com has some good explanations on turbos and superchargers.