Refer to figure.
Learning Objective 021.10.10.01.04: Describe the function and the principle of operation of the following main components of a turbocharger: turbine; compressor; waste gate; waste-gate actuator.

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. 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 puts a lot of strain on the engine.

A turbocharger works by using the speed and pressure of the exhaust gasses to spin a turbine, which then uses that energy to spin the compressor, which compresses the fresh intake air ready for entry into the cylinders. We cannot allow the turbocharger to run wild though, as too much pressure (MAP) can cause the engine to "overboost", and even fail quite spectacularly.

The way we regulate the turbocharger is by varying how much air gets to the turbine, this is via a wastegate system. The exhaust gasses can go down two different tubes, one is a standard exhaust, straight out of the aircraft (bypassing the turbine). The other is a tube to the turbine, to spin up the turbocharger.

The wastegate is a butterfly valve that can block the exhaust tube, so when it is closed, all the exhaust gasses go to the turbine, making the compressor/turbine spin fastest.

When the engine reaches its maximum MAP, the oil system which regulates the wastegate begins to open it, letting some of the "waste" air straight out of the aircraft, so it does not run the turbine, which might overboost the engine.