Shrimp cocktail is simply cooked, peeled, chilled shrimp, served with a tomato-based, usually spicy sauce. You can cook and peel your own shrimp, which has the advantage of leaving you with shrimp shells with which to make shrimp stock, but it's undeniably convenient to just thaw out frozen pre-cooked shrimp.
I find that the sauce that you can buy at the store isn't nearly spicy enough. It's not hard at all to make a much better sauce with easily available ingredients:
Shrimp cocktail sauce:
1/2 cup ketchup
1 teaspoon chili powder, or more if you like
1/2 teaspoon cayenne pepper
1 teaspoon finely minced onion, or 1/2 teaspoon onion powder
1/2 teaspoon salt
1 teaspoon to 1 tablespoon Worchestershire sauce
1 tablespoon prepared horseradish, or more if you like (and I do)
juice of 1 lemon
Just mix the above ingredients and chill it in the refrigerator if you have the time. Serve with the chilled shrimp (the above recipe makes enough sauce for about 1 pound of shrimp).
Prepared horseradish is the only thing which isn't a typical item, but if you follow the link, you'll find a writeup on how to prepare it from the raw root, which can be found at almost any grocery store, and once prepared, it lasts for months and months. And there's all kinds of other things which homemade horseradish makes a wonderful accompaniment to. Mmm...