Monitors are speakers where the 'audience' is a musician (or teccy) who needs to hear something to do their job.
Most important for vocalists. To sing in tune, you have to be able to hear yourself. For choirs in churches they just listen to the noise coming from their own mouths. On-stage, however is a different environment, with backline, drums and anything upto a few hundred K of PA and a few hundred thousand screaming audience, hearing your own voice needs a little extra 'amplification'.
Quite a cool job is to be monitor engineer; you'll certainly get to know your desk and feedback frequencies really quickly, because all the speakers are pointed at the singer's ears, about 4 inches from the microphones.
Monitor speakers are called wedges, because, well, duh! they're wedge shaped.
Remember the 'Polar patterns' of the mic's you're using when positioning the wedges. And to ring-out thoroughly beforehand ( find feedback frequencies and eliminate them), but don't be too severe in cutting!
These monitors are bastard complicated, if you're playing with these, especially the sort where the artist has their own mix-pack, then you don't need my guide to help you (I hope!).
Using your extreme knowledge, position approx 140kWRMS of PA in a concentric pattern around the drummer. This is called "DRUM FILL" and it's aim is to implode the head of the drummer. Anything below this level of sound is "rubbish" and you will be a "Bastard Wanker" and all war and famine in the world will be your fault. According to the drummer.
Despite common misconception, orchestral groups do use monitors occasionally, however, not huge 15" and compression drivers, but normally ickle tiny, nasty-sounding things from BOSE and JBL, but cellists and the like don't seem to notice.