This refers to the size of the chainrings on the front of the bike, which directly reflects on your gearing.
"Double" means that there are two front chainrings (gears) and is advantageous in that it is lighter and much easier and simpler to work with.
"Triple" means that there are three front chainrings (gears) and is advantageous in that it gives you more gearing options.
Standard chainrings are larger (53/39 tooth), and with them, you will have more pedal power benefiting you on descents and flat ground.
Compact chainrings will be smaller (50/34 tooth) giving you an easier ratio needed to climb hills.