Ok, my $0.02:
Super or turbo charging basically accomplish the same goal. The -charger feeds compressed air into the intake of the engine. A stock -charged car may run anywhere between 4 and 14 psi of boost.
Superchargers are typically belt driven, just like any other accessory on the front of the engine. Bascially think of it as a "fan" that blows air into the engine at high pressure. In reality, it's not a fan - usually a screw or other type of compressor. It is not "easy" to generate hundreds of cubic feet of air at high pressures. Ever tried to blow up a balloon with a fan? Not happening. The big advntage of supercharging instead of turbocharging is that there is "no lag" - as long as the engine is spinning, it is turning the supercharger at the same rate. The faster the engine spins, the more air the supercharger pumps in.
Turbochargers are funky devices that attach to both the intake AND the exhaust. Think of them as a double-bladed fan. One fan sits in the exhaust pipe, the other in the intake. When your engine spits out exhaust, it spins that fan, which in turn causes the intake fan to spin as well. So the more exhaust, the more pressure it can build up at the intake side. This is why you get "Turbo lag" - the exhaust gasses have to have enough velocity to get the turbocharger spinning. Turbo turbines (they literally are fans) spin at very very high (50,000?) rpms - which is what allows them to generate pressure. As the exhaust fan spins faster, it make the intake fan spin faster - generating more boost and more power.
Both processes generate HOT air - that's physics, compressing a gas creates heat. When you hear about "intercooled" engines (usually in ref to turbos, not superchargers) that's basically an air-air heat exhanger (radiator) which allows the heated air to cool down before entering the engine. Usually superchargers are bolted right to the top of the engine, so no room to put an intercooler in.
The downside with either is that you've added complexity to the engine, so more things to break etc. Also your fuel management / engine computer systems get a little more complex to deal with the implications of adding all that extra air. (which means you add extra fuel, which means your little engine acts like a bigger one). It is usually easier to modify turbo or supercharged engines, because they'll have a stock "boost" level (say, 10psi) at which the system starts dumping off excess pressure. If (through some aftermarket gizmo) you raise your boost from 10 psi to 14psi, you could quite realistically see a 20-30% increase in horsepower. That's very difficult to achieve in a non-charged (naturally aspirated) engine. On some cars that is a REALLY easy modification - $5 in parts and you're there. However, you have to make sure that the engine computer, etc, etc can handle it - the biggest drawback is that you can run out of fuel before you run out of air - so you'd have to increase the size of your fuel injectors, etc.
PS: the "upside" to turbo lag is that typically turbocharged cars will hit a certain RPM when the turbos will really "spool up" and you'll feel them come on strong. In most, this would be in the 3000-4000 rpm range - suddenly you'll get a big spike in engine output. You all watched knight rider, right? hahaha. Anyway, it's an entirely different feeling from a high power V8, etc which make so much torque down low in the RPM range. Most "quick" cars (and even some not so quick) will keep up with me from a light until the other side of the intersection - around there, the turbos kick in, and I just walk away...