The key to understanding a turbocharger is understanding volumetric efficiency. This is a measure of how effectively an engine is breathing, so force feeding it air makes it more powerful, which is what a turbocharger does.
If an engine could completely fill the cylinder with air then it would have 100 percent volumetric efficiency, but it's rare for anything other than a full-on race engine to do this. With only normal air pressure to push air into the cylinder, highly tuned naturally aspirated engines can use resonance in the air inlets and valve overlap to increase the amount of air getting into the engine, but that only goes so far.
What is needed is something that increases the pressure of the air getting into the engine, and that means a turbocharger. A turbo relies on exhaust gases to spin a compressor wheel that forces air into the engine.
How does a turbocharger work?
A turbo is essentially two fans connected to each other via a shaft. One of the fans sits in the exhaust stream and has fins that are spun by the exhaust gas flowing past. Through the shaft and on the other side there is another fan that pumps air from the air intake to the engine. The air gets pumped through the compressor wheel/fan and is forced towards the outside of the turbo through centrifugal action and then into the engine.
But the compressor needs time to reach optimal operating speed, so there is a varying amount of lag between the throttle opening and the turbo producing power, depending on what size of turbocharger used. Turbos generally don't work all that well at low rpm, because the compressor needs to be spinning quickly to pump the air into the engine.
Blow off Valve
This is all great while the engine is producing power, but when you lift off the throttle to slow down or change gear there's still pressure in the intake system, which is why you need a valve to relieve the pressure so nothing bursts. The blow off valve is popular because you can vent it externally and make the whoosh sound you hear so from so many turbocharged cars when they change gear.
How does a Blow Off Valve Work?
Most modern turbocharged cars come standard with factory fitted BOV's and don't really need aftermarket BOV's. The factory setup is plumbed back into the air intake after the airflow meter and before the turbo intake. There are two reasons for this, noise control and most importantly, if the vehicle is fitted with an airflow meter the air being vented by the BOV has been measured by the ECU. If you then vent this air out of the system the Air/Fuel ratio becomes wrong and the engine becomes out of tune for a period of time and it will not meet emission tests.
How does a wastegate work?
A wastegate is a different thing entirely, as this is used to stop the turbo from producing too much pressure as it starts spinning too fast. This is usually activated through a spill valve in the exhaust system prior to the turbine wheel and is operated by a spring relief valve that regulates the speed of the turbocharger rotating assembly and with that, the boost pressure.
This ensures that your boost doesn't overpressurize the engine and result in engine cylinder detonation, which is when the fuel in the cylinders explodes under heat and pressure rather than being burnt in a controlled wave front at the correct time in the compression stroke.