The Booster is an after market item made by several companies. It can be mounted in several places. Directions say to put it somewhere that will get cooling air. I have mine on the fairing frame. Some riders put it under the tank. It's really too big to fit under the front cover.
This seems to be the most popular. There are others, some cheaper maybe. Or you can also build your own. Since they can be attached to the bike in almost any spot a lot of times new riders actually have a booster and don't know it (happened here a week ago, or so). The easy way to tell if you have one is look at your wiring. The points wire goes to the condenser and the right coil wire goes to the condenser. If this is how the bike is wired it does not have a booster. If there is a booster a wire from it will go to the coil.
There are other systems also that replace more of the OEM system. They may even have an internal electronic advance, eliminating the mechanical advance of the bike. The advantage of the booster is that it keeps all of the OEM system and uses the ignition points as a simple switch that only has to carry normal battery voltage. In the OEM set up the points carry 20,000 Volts and this is what usually wears them out. By eliminating this high Voltage at the points they can last a long time. With a Booster the mechanical advance is kept and the purest amongst us appreciate that.
Since it is a man made item the Booster can break like anything else. But to revert to the OEM ignition from a Booster involves only moving a couple of wires. Biggest advantage of all. All the original parts are still attached to the bike. We've had long discussions about some more complicated electronic ignition systems and the riders who buy them swear they will pack the points and points plate till that eventful day when it is needed. I did this when I had a Dyna III system. After less than two years of packing these pieces in a plastic bag (this insures the parts will be constantly wet and never dry out) the parts were not usable.
The usual reason for getting a Dyna III is because there is a ghost timing image and the firing of the two cylinders can be controlled in the Dyna III by two separate magnetic pick ups. This is hard to set up but it can work. After I blew the Dyna III by trying to balance carbs with out using shorting rods but just lifting the plug wires I had to replace all the stock parts that were rusted up in my plastic bag.
There is a way to fix a Ghost Timing Image with out having to eliminate the stock ignition system. It's a little more complicated but doable by the Home Mechanic. I got rid of the Ghost, Double Timing Image this way.
I'm a fan of keeping the stock ignition system with it's mechanical advance and adding a Booster.