There are hundreds of shrimp species which can be divided into two broad classifications - warm-water and cold-water. Generally, the colder the water, the smaller and more succulent the shrimp. In North America, shrimp are marketed by size, expressed as an average number per pound. The general size categories are: colossal (10 or less per pound), jumbo (11-15), extra-large (16-20), large (21-30), medium (31-35), small (36-45) and miniature (about 100). (In North America very large shrimp are sometimes erroneously called prawns, which are actually a different species altogether.) In general, one pound of whole, raw shrimp yields half to three-quarters of a pound of cooked meat.
Shrimp come in many colours including reddish- to light brown, pink, deep red, grayish-white, yellow, gray-green and dark green; some have colour striations or mottling on their shells. However, when cooked, most shells change colour - say, from pale pink to bright red or from red to black.
If you're lucky enough to live somewhere where you can get fresh shrimp, by all means do, and count yourself lucky. Here in Toronto all I can get is frozen ones, and that's too bad, because freezing compromises their texture and taste. Be sure not to thaw frozen shrimp until you're ready to use them; they will spoil quickly once thawed. See A simple but better way to defrost shrimp for tips.
I usually buy unshelled shrimp because they are cheaper than shelled. To peel, tear off the legs and then remove the shell. (Boil the shells in water for a quick shrimp stock.) Whether or not you devein is a matter of personal preference, but the "vein" is an intestinal tract, so I usually do. Just make a cut along the back of the shrimp and remove the thin black thread that may be present.*
By the way, my shrimp of choice are Thai tiger shrimp, shelled but headless. I buy them in Chinatown where they are cheap and have come directly from Thailand, rather than sitting in a supermarket freezer for months.
*Update: I've just discovered How to peel and devein shrimp, which has detailed directions for a few ways of performing this feat. And also A simple but better way to defrost shrimp.