Turbochargers harness some of an engine's wasted energy to compress the air going into that engine - thereby increasing the output of the engine. Perhaps you can sense something cyclic about the previous sentence. That's because a turbocharger is by its very nature, cyclic. Use of a turbocharger can increase the output of an engine by 30 percent or more without a significant increase in the weight of the vehicle.
At the end of combustion, as the piston is traveling back up the cylinder, the exhaust valve(s) opens and hot air is forced out of the cylinder. This exhaust is wasted in a naturally aspirated engine, simply being released into the atmosphere. A turbocharger, however, recycles this energy. In a turbocharged setup, the exhaust gases are forced through the turbine of the turbocharger. This causes the turbine wheel to spin.
The turbine wheel is directly attached to another wheel in the compressor of the turbo. On this side of the turbo, often referred to as the compressor or the “cold” side, the spinning compressor wheel compresses the intake air. This pressurized air is then forced into the engine. Since pressurized air has more oxygen per unit volume (PV = nRT), there is more opportunity for combustion. This additional oxygen can either accompanied by additional fuel for increased power or in some diesel applications, the advantage is in the excess oxygen which allows for cleaner and more complete burning of the fuel, resulting in reduced emissions and increased efficiency.
When air is compressed so that more air can fit in the same volume, its temperature is increased. This is compounded by the fact that the turbocharger is not 100% efficient and generates heat when compressing the air. Since the air is hotter, it is less dense, which means less oxygen is in the cylinder than if the air were cooler. This explains why a 14.7 psi turbocharger gauge pressure will not double the power of an engine. The heat also explains why an engine under boost is more prone to knocking (predetonation or auto-detonation). In order to cool the charged air, an intercooler or water/alcohol injection is often used.
Because a turbo is a cyclic device (the more exhaust it gets the more air it can compress which makes even more exhaust), left unchecked it would create a huge amount of boost destroying your engine very quickly.
In order to keep the boost manageable there needs to be a way to limit the amount of exhaust gas that turns the turbine. This is what a wastegate does. The wastegate is a valve that opens at a particular pressure and allows exhaust gases to circumvent the turbo.
Wastegates are usually integrated with the turbo, but many applications use an external wastegate which is totally separate from the turbo (this is the setup on the Turbo Esprit.) Wastegates have a spring that is used to set the pressure that the wastegate will open at, therefore setting the “boost” of the turbo. The non-HCI Esprit's wastegate is set to 8psi.
If you have an interest in changing the pressure at which the wastegate opens you must either swap the wastegate spring, bleed of some boost so the wastegate only “sees” a percentage of the actual boost pressure or “hide” the pressure from the wastegate using a solenoid. The third method is the preferable way to adjust your boost (in fact a number of turbo charged cars are engineered from the factory to use this method.)
By outfitting your car with an electronic boost controller you can adjust your boost level from the driver's seat (you can easily run more boost at the track with a higher octane, or on colder days.) With some of the more intelligent boost controllers you can make your boost level RPM dependent (maybe 12 psi at 3000 RPM, but only 8 psi at 6000 RPM.)