Brownies can be fudgy or cakey, depending on their density. Fudgy brownies are compact, moist and rich, while cakey brownies have a lighter crumb that more closely resembles chocolate cake.
The difference comes from recipe proportions and preparation. More fat from butter or chocolate, less flour and little or no leavening generally encourage a dense, fudgy result. More flour, beaten eggs or baking powder can create greater lift and a cakier texture. Baking time also matters: a slightly underbaked centre stays moister.
Brownies developed in the United States in the late 19th century. Early recipes were often closer to cake because they creamed sugar and butter; later recipes became richer and fudgier. The 1893 Palmer House version is one famous early brownie tradition, although the name was attached to it later.
Chewy is a separate description that can overlap either style, so it is not the best opposite of fudgy in this question. Spongy and flaky describe other baked textures rather than the standard brownie pairing.