An eatery is a place where you go to eat — a restaurant, a café, a little local place. It is eat + -ery, the same kind of ending you hear in bakery: “a place connected with this.” The word is informal. Newspapers and travel writing like it (“a cheap eatery near the station”); a fancy restaurant is less likely to call itself an eatery. Though actually, it is not the everyday word most people say when they are hungry — they say restaurant, café, or the name of the place. Eatery has a slightly breezy, American, “somewhere to grab food” feel.