Shrimp and prawns are crustaceans, related to lobster, crab, crayfish, and krill. Prawns are not shrimp, but are generally indistinguishable in culinary usage; see the recipes listed below for details.
To remove the shells, start by pinching away the leg-like gill structure. Then split the shell from there, pulling it off the rest of the way. You may wish to remove the "vein" on the back, which is actually the digestive tract. If the shrimp are already deveined, they will be split along the back, and a good way to peel them is to simply squeeze forefinger and thumb against the tail. This will push the "meat" out and allow you to easily pluck off the remnants of the shell. Of course, this results in tail-less shrimp.
The easiest way to cook shrimp is by boiling. Simply place them into a pot of boiling water and cook them until they change color, generally becoming pink. You can serve the shrimp with lemon juice, ketchup, a mixture of both, or a fancier cocktail sauce.
A more difficult and dangerous deep fat frying procedure can be used to produce fried shrimp. There are several types including Southern fried shrimp (non-puffy batter containing cornmeal) or Chinese fried shrimp (puffy cornmeal-free batter, as used in westernized Chinese cooking), and tempura (thin and sometimes flakey Japanese style).
The shells and tails from peeled shrimp can be used to make a stock. Simply place the shells and tails in a pot with a small amount of water, bring to a boil, then cover and simmer while you prepare the rest of the dish. This stock can then be used to flavour the dish, and/or used as a component of the cooking liquid to make shrimp-flavoured rice.