Fresh sausage - like breakfast links, sausage patties, Italian sausage, bratwurst, and Mexican chorizo - is made from chopped, ground, or even pureed uncooked meat. It must be kept refrigerated or frozen until used.
Prick the skins lightly before cooking to avoid exploding sausages. Cook fresh sausages thoroughly, until no hint of pink thinks about hanging around in the center of the sausage.
Fresh sausages can be grilled, pan-fried, or broiled. Some types of fresh sausage - particularly "white sausages" made with veal - are traditionally poached or steamed. Fresh sausages can also be cut up or removed from their casing and crumbled in order to be sauteed and used in other dishes.
Pre-cooked sausages - like hot dogs, frankfurters, bologna, mortadella, and many German-style "wursts" (although not all, be sure to ask your butcher or sausage-maker whether a sausage is fresh or pre-cooked if you have any doubts) - are usually made with smooth, even pureed fillings. Sometimes the fillings are partially cooked before being stuffed into the casing, but in all cases the sausage is cooked after filling.
You still want to cook these sausages - or at least heat them up to bring out their best flavor. They can be grilled, broiled, pan-seared, or cut up and added to other dishes. You don't need to worry about cooking them through or to temperature, though, since you're not dealing with raw pork.
Smoked sausages like andouille and kielbasa are, as their name suggests, smoked to cook them. They are hung in a smoker or smoke house where a cool fire is tended to burn slowly and to produce a copious amount of smoke that cooks, flavors, and preserves the sausages (or other meats being smoked).
Smoked sausage can be eaten as-is, heated, or cut up and used in other dishes. Smoked sausages, like their pre-cooked friends, are a staple of deli counters and sandwiches.
Cured sausages are what Italians call salumi and the French call chacuterie - they are sausages made fresh and then salted and air-dried for weeks or months depending on the type. The meat gets "cooked" by the salt and air (and, in a way, time). Spanish chorizo, coppa, and Genoa salami are just a few examples of cured sausages.
Slice cured sausages as thinly as possible and serve at room temperature for a great appetizer or snack.