Refer to figures.
Learning Objective 021.10.10.01.10: Explain the purpose of a supercharger and the basic differences from a turbocharger.

Piston Engines can have their power increased in three main ways, by increasing the size of the engine (displacement), increasing the RPM, or increasing the pressure of the air/fuel mixture (called the Manifold Absolute Pressure, MAP). This last case is why we might use a turbocharger in our car engines, so we can have a small, efficient engine, with extra power available from a turbocharger (which compresses the intake air) when needed.

Turbochargers are arguably even more useful in aircraft, where piston engines become less powerful in the thinner air at higher altitudes. This is where turbochargers can compress that air, to make the engine operate with the same power output it would do at a much lower altitude. We call this an "altitude boosted" turbocharger. They are more useful to us by keeping the same power up to higher altitudes, rather than giving us lots more power on the ground, which would put a lot of strain on the engine, which would be a "ground boosted" turbocharger.

In reality, a turbocharger is a particular type of supercharger. All superchargers compress the fresh intake air so more fuel/air mixture can be burnt in each power stroke, thereby making the engine more powerful (or allowing it to maintain its power to higher altitudes). Supercharger is technically the umbrella term that covers all such devices, but in reality, most people refer to power augmentation devices as follows.

Supercharger: An engine driven compressor which allows an increase in MAP.
Turbocharger: A device where the excess heat and speed of the exhaust gasses drive a turbine which drives the compressor, allowing for increased MAP.

There are various reasons for using each; superchargers do not suffer from turbo-lag, but do take power directly away from the crankshaft, whereas turbochargers are more efficient by taking energy away from the otherwise wasted exhaust gasses. In aircraft, turbochargers are far more widely used, but the learning objectives do ask us to understand the key differences between the two.

Turbochargers do cause a small power loss that is not often talked about, and that is due to exhaust back pressure. Essentially, in an non-turbocharged engine, the exhaust gasses are allowed to flow freely from the cylinders during the exhaust stroke, but when a turbocharger is added (and the waste-gate is closed, meaning that all air goes through the turbine), there is some back pressure from the more restrictive flow through the turbine. This means that exhaust gasses are not removed as effectively, and the engine does lose out on a small amount of power. It is not a lot though, as the extra MAP means that the power increases massively for this small loss.

On normally aspirated and supercharged (not turbo-) engines, though, the exhaust gasses are allowed to leave the engine without restriction, and technically, the higher we fly, the lower the ambient pressure, so the lower the exhaust back pressure. This is a very small gain of course, but it can make a difference, especially when a supercharger makes the engine perform like it is at a low altitude, but with even better exhaust gas removal.