When you cook a dish au gratin, you sprinkle it with breadcrumbs and brown the top in a hot oven. Some au gratin recipes also call for melted, browned cheese.
If you make potatoes au gratin, you might slice them thinly, layer them in a baking dish with cheese, top with fresh bread crumbs, and bake the whole thing in an oven for an hour. There are many variations on casseroles that can be prepared au gratin and made even more delicious. The phrase au gratin literally means "by grating" in French, or "with a crust," from the verb gratter, "to scrape, scratch, or grate."