First: The "how" and the "why"
Any type of forced induction system has one purpose: increase the volume of air and fuel going into the engine's cylinders, resulting in higher compression. Although there are different ways of accomplishing this, the standard procedure is to increase the volume of air being sucked, or pushed, through the intake manifold. The standard procedure involves spinning turbines or rotors (imagine a fan). Doesn't the cylinder already compress the air-fuel mixture as part of the combustion cycle? Yes it does, and that's what we want. Forced induction just helps the engine compress the mixture more than it can on its own. More compression is good because it results in a bigger bang during ignition. The outcome is a more powerful, yet not bigger, engine. That about sums up the challenge today's designers face thanks to economic changes and unpredictable oil prices.
A supercharger is the key to big power and acceleration from a standing start. It's essential for tire-shredding burnouts, blistering 1/4-mile times and brute force acceleration. Most commonly, a supercharger is a chamber containing rotating screw- or cam-like shafts that repeatedly compress air as they spin. In this case, the rotation is provided by a direct mechanical connection to the engine's crankshaft. Usually a belt and pulley are used to transfer motion between the two shafts. This does consume some of the base engine's power, but there will still be a net gain in power at the wheels.
A turbocharger is, well, a supercharger. There are a couple of main differences though. First of all, a turbocharger is turned by the flow of the engine's exhaust. An impeller is spun by the exhaust pressure, and turns a second impeller which forces air back towards the intake. The whole process results in sort of a chain reaction. The pressure will build up in the exhaust at first, as it gets the turbo spinning. Once it's going though, the whole system builds on itself quite quickly. A turbocharger can produce tons of power, but is generally associated with lag at low RPM. Modern automotive turbochargers are designed to minimize this, resulting in much more responsive engines.
Chances are, turbocharged, four-cylinder engines are going to become very popular in the U.S. very soon. It's already happening. Everyone wants to know about MPGs, but no one wants to give up power. A turbocharger is the key to making small engines feel bigger, and bringing some fun to the economy car scene. Modern muscle cars like the Corvette ZR1 and 2011 Shelby GT350 will continue to count on superchargers for horsepower numbers more than half-way to 1000. I'm all for it. Although cars have become a daily necessity for most of us, they should still be fun.