To classify organisms, we use the five-kingdom system of classification. Obviously, the crayfish is not an amoeba-like organism or a single-cell organism. It is clearly not a plant, because it does not manufacture its own food, has a nerve cord and can move. Therefore it must be an animal. It is actually relatively complex as animals go. I know its classification to phylum level: Kingdom: Animalia - Animals.
Subkindgom: Metazoa - Animals with a digestive tract.
Phylum: Chordata - Animals with a neural tube. The crayfish, as a crustacean, is not a vertebrate, but it is still quite a complex animal.