If you order mutton at a restaurant, you'll be served a kind of meat that comes from an adult sheep. It's not, however, very common to see mutton on a menu.
Use mutton when you're talking about eating the flesh of a sheep. In the United States, it's more common to dine on lamb, which is meat that comes from a baby sheep, although mutton is a popular dish in other parts of the world. Mutton has been in use since the thirteenth century, from an Old French root word, moton, "mutton, ram, or sheep." In the 1860's, long sideburns called mutton chops were a popular style of facial hair.