The concept behind the Minion method is simple. Place a small number of lit coals on top of a pile of unlit briquettes and control your temperature by adjusting your vent(s). The lit coals will ignite the unlit coals slowly, allowing you to cook for long periods of time without having to add new coals as well as controlling your temperature since it is easier to bring a fire up to a certain temperature rather than try to bring it down. You really only want to use this method for slow and low cooking (225-300 degrees), not for 325-350 degree cooks.
One possible con to this method is that some say you should never add unlit coals to your fire becuase of the fumes that get released when the briquettes ignite. While this is true, I have not noticed any foul taste on any of the meat that I have smoked.
Depending on the amount of unlit coals you start with, you are supposed to be able to get up to 18 hours of heat by using this method. I have not done anything more than 9 hours and I had to reload coals three times in that period using this method. I can typically go up to 3 hours without having to add any new coals if I use the Minion Method. I only start with one full chimney starter of unlit coals and 3/4 to 1 chimney starter of lit lump charcoal. I'm sure I could go longer if I started with more.