You can get an idea of what each of these looks like by going to the Performance website (or any other online bike shop).

Triples are typically used on touring bikes that may be heavily loaded and by older folks like me with tired knees who need easier gears for climbing hills. A triple these days often has the standard chain rings 53 and 39, plus a "granny" ring which is usually a 30 tooth ring for the extra low gears. Road triples usually use a 130 mm bolt circle for the two big rings (just like a standard double) and a smaller bolt circle for the granny ring (I don't recall the size off hand).

Standard double cranks use a 130 mm bolt circle. This restricts the small ring to being no smaller than 38 teeth. Most standard doubles today ship with 53 and 39 tooth rings.

A compact crank is one that takes 2 chain rings, but uses a 110 mm bolt circle so that it can take rings as small as 33 teeth, although most compacts ship with either a 34 or 36 tooth small ring. Almost all compacts also ship with a 50 tooth big ring. Thus, compared to the standard double, you give up 3 teeth on the big ring. You can do the math yourself to determine how this affects gear inches, but for me, for example, I never pedal fast enough to need more than a 50x12 gear (if I am going faster than 35 MPH, I will always coast ;-)

The advantage of a triple is that it provides the widest range of gears and can be set up so that the middle ring covers a wide range of useful gears, thus requiring use of the big or small rings only occassionally and cutting down on front shifts. The disadvantages of a triple are (1) a bit of added weight; (2) many triples entail a larger Q factor (your feet are farther apart to make room for the extra gear) which some find less comfortable; (3) front shifting is a bit less snappy and precise because of the extra ring and associated hardware compromises; and (4) bike snobs will often sneer at riders using a triple (they feel such riders are not "man enough" to ride a normal double.

The advantage of a compact is that it provides a gear range almost as wide as that of a triple without the small added weight penalty or the reduction in shift performance associated with a triple. Also, for those who already own bikes with standard doubles, the cost of switching to a compact is less than the cost of a triple. Converting to a triple often requires a new front brake/shift lever and almost always requires a new front deraileur and may also require a new bottom bracket (for those without the new outboard BBs). Converting to a compact typically does not require a new front deraileur and does not require a new shifter.

The disadvantages of a compact are: (1) there is a larger gap between the number of teeth on the rings than on a standard double, this leads to two issues (a) you shift rings much more often with a compact than with a standard double and (b) you frequently need to shift both front and rear simultaneously to make the gear switch comfortable; (2) some compacts shift less well than standard doubles, but others are just fine (a compact that shifts really well is the Shimano R700); (3) triples typically give one gear lower and one gear higher than a compact, so if you want/need the extra gear on the top or bottom, the compact won't get you there; (4) bike snobs will often sneer at riders using a compact (they feel such riders are not "man enough" to ride a normal double.

Hope this helps,
Ira