What is a turbo?
Simply, a turbo is an exhaust-driven compressor. Imagine a small shaft about the size and length of a new pencil. Now rigidly attach a pinwheel to each end of the pencil. One pinwheel (called the turbine) is placed in the path of the exhaust gases which are exiting the engine. These gasses are 'caught' in the turbine, causing it to spin. This in turn spins the whole shaft, along with the pinwheel on the other end (called the compressor). The compressor is placed in the intake air's path; once it begins spinning, it actually compresses the air on its way into the engine.
Why is this beneficial? Well, normally aspirated engines have to work to draw in their intake air. In other words, as the intake valves open, the piston's downward movement creates a vacuum which 'sucks in' some air through the intake system. Ideally, the piston's movement would suck in 100% of the air that could fill the combustion chamber. In the real world this is not the case; the typical engine will draw in only about 80% of the total volume of the combustion chamber. There are many reasons for this--intake restrictions, valve timing, camshaft design, and much more.
Now imagine that the engine mentioned above has a turbocharger. When the turbo compresses the air it builds up pressure in the intake manifold. So now when the intake valves open, air is actually forced into the combustion chamber. (This is one reason why turbocharged engines are sometimes referred to as 'forced-induction' engines.) As you might imagine, this allows for more air to fill the chamber; the amount will still not equal 100% of the entire volume but nonetheless it is closer to this elusive goal.
Okay, so now we have more air entering the engine. To benefit from this, we need more fuel to match. On computerized cars, various sensors will see this amount of boost pressure and increase the amount of fuel accordingly. Now that we have more fuel entering the engine, more power is made. (Yes, when you get right down to it, the only way to make more power--on any engine--is to burn more fuel.)
How do turbochargers and superchargers differ?
While they perform the same function, turbochargers and superchargers go about it in completely different ways. As has already been mentioned, a turbo is driven by the exhaust gasses which are already being expelled from the engine. So, in effect, turbos add 'free' power since their compression is created by what was already discarded.
Superchargers, however, are different: they are belt-driven. They feature a pulley whose belt is directly attached to the crankshaft, this allowing them to spin in direct proportion to the engine itself. This offers the tremendous benefit of avoiding any lag ; at least some boost is available any time the driver decides to hit wide open throttle (WOT). The drawback to a supercharger, however, is that they take power to make power. The overall result is more power than there would be without the supercharger; it's just that they aren't as efficient as a turbocharger from an energy standpoint.
