Turbos use exhaust pressure and volume to drive a turbine connected directly to the end of the manifold, where all the mains come together. The exhaust then flows through the exhaust sysetm, but when too much exhaust fills the manifold, it is evacuated through a waste gate as not to damage the turbo. While the turbine form the exhaust side is spinning, it's in turn spinning another turbine from the inlet side. Air is sucked in through the intake and is compressed in the turbo. The air is pushed through pipes into an intercooler which cools the hot air which was just passed through the turbo inlet. Air then travels into the engine. The faster all this happens, the faster the turbines will spin and inturn turn out more air, producing more boost and power. When the throttle body closes (taking your foot off the gas), the build up of pressure will be released through a by pass valve, also known as a blow off valve, so the build up doesn't back track and hurt the turbo.
Supercharger is less complicated. A belt turned by the crank, turns the blower or supercharger which rotates the turbines and compresses air which is tossed into the engine. The higher rpms will = more boost and power. A general misconception with these is that they don't make boost throughout all the rpms. With small superchargers, boost will stop being made after a certain rpm because of the small size of the turbine. The bigger the supercharger, the less boost will be made at lower rpms because the air will leak out due to the size of the supercharger. So all superchargers dont run throughout all rpms.
Little more extensive info for ya