What is the meaning of Au Gratin in cooking?

Au gratin is a culinary style of topping a dish with a cream sauce or creamy cheese sauce and herbed bread crumbs. Au gratin does not necessarily mix the sauce all the way through a dish but most often it will mix that way, often by design.

For au gratin potatoes (pommes de terre au gratin) Slice and layer potatoes into a casserole with a few pats of butter interspersed, make a thin bechamel sauce and melt a cheese such as Cantal or something similar into the bechamel. Pour the sauce over the potatoes, sprinkle with herbed breadcrumbs or croutons. Bake in a medium oven for 45 minutes until the potatoes are softened.

Bechamel is made by melting a half cup of butter and sauteeing about 1/2 onion finely diced. when the onions are translucent add 1/4 cup flour and blend it until smooth then add 1 to 1 1/2 cup milk or cream and a pinch of nutmeg, stir until thickened. To thin it add milk, to thicken it add less milk or simmer it longer without burning it.
Melting cheese into this will make a cheese sauce. thinning this and adding various ingredients will make various cream sauces or soups.