Mono - Has more stretch and shock absorption, and doesn't sink rapidly. Good for crankbaits, lipless crankbaits and some topwaters like a Pop-R or walking baits like a Spook to help keep the action right during the retrieve.
Fluorocarbon - Allows more light through the line so it is clearer underwater and less visible to fish, especially in clear water. More dense, so it sinks and isn't great for topwaters (like the ones mentioned above) but is OK for buzzbaits. Less stretch and, with heavier sizes (14- to 20-pound test) is used for Texas-rigged worms or plastics and jigs around cover. Also OK for crankbaits.
Co-Polymer - A mix of mono and fluoro, with qualities of both. A good all-around line.
Braid - Zero stretch. Very sensitive. Great for flipping plastics and jigs. Because of the small threads woven together to make the line, the line diameter will be small compared to the pound-test size. For example, 30-pound braid will have the size (diameter) of 10-pound mono ... 10-pound braid will be about the size of 2- or 4-pound mono ... 65-pound braid, for heavy-duty flipping in weeds and cover, will be about the size of 25- or 30-pound mono.
I use braid for topwater frogs, jigs, worms, spinnerbaits, buzzbaits and have played with it with crankbaits, although with no stretch you'll either get 'em in the boat or rip the bait from the fish's mouth if they don't have it good.
Braid CONS - Bad backlashes can be tough to pick out of the reel spool. If the line isn't spooled on tightly, when you set the hook it can dig into itself on the spool. Definitely a learning curve.
If you want to fish saltwater, I'd go with fluorocarbon or co-polymer.
If you're pond/lake fishing, 10-, 12- and 14-pound test should be fine. You could use the 12 in salt and probably be OK, too.
