Triple = 3 chainrings, usually 50/39/30 for a road bike
Compact = 2 chainrings, usually 50/34
Both of them offer lower gear ratios than standard (53/39) cranksets and are more appropriate for the average rider that doesn't race.
Triple is best if you have lots of hills or have strength/weight issues. Compacts can be annoying due to the 16 tooth difference in the two rings. Triples get a bad rap for dorkiness, excess weight, and lower quality shifting. I think that's all a load of do-do propigated by certain groups of riders. A quality road triple with good shifters and derailleurs will shift smoothly and quickly when assembled with care by a good mechanic.
