The primary control of boost in a turbocharged system will be the wastegate. The purpose of the wastegate is to bypass the turbochargers turbine, reducing its ability to make boost. (For more information on how the turbocharger works, see our article on turbochargers.)
The wastegate will have a pre-set spring pressure. This can be adjusted by changing the spring on most, and also by varying the preload of that spring in some others. Outside of that, a boost controller can be used.
Boost controllers come in a variety of different types, however, there are only two methods for raising the delivered boost; you must either reduce the pressure on the bottom port of the wastegate, or add pressure to the top port (if available) on the wastegate.
For reducing the pressure on the lower port on the wastegate, there are two popular methods. One is a “controlled bleed”. This is found in mechanical boost controllers that are “needle type” valves, as well as the method used by some electronic boost controllers by pulsing a solenoid to control the amount that is bled off. The other is to artificially hide the boost signal from the wastegate until an alternate target boost is reached, as is done mechanically with a “ball and spring check valve”. Some electronic boost controllers use this method as well. Some electronic controllers also will blend between the two, hiding the boost from the wastegate until a target pressure is reached, and then acting like a controlled bleed once that target is reached.
Both of the above are common implementations using only the lower port on the wastegate. The top port is also very useful in that you can leave the bottom port unmolested, and add pressure to the top port only when more boost is desired. Usually this is only something that electronic boost controllers will use, but some mechanical boost controllers also take advantage of this port.
On electronic vs manual controllers, it strongly depends on which versus which. In general, electronic controllers have an advantage of being able to react in a non-purely-mechanical means. So if you have a mechanical system that becomes inefficient and tapers at higher RPM, some electronic systems give you the ability to compensate and command more boost at higher RPMs. Mechanical systems tend to only be able to translate the natural boost curve up, but not drastically alter its shape.