Six classes have been discovered to date but Axie Infinity suggests there will be at least three more.
Classes are unique in that they apply both to the Axie itself as well as body Parts. For example, it is possible to have an Axie classified as "Beast" with Bird, Bug, or Plant features but without any Beast parts! (see Axie 4098)
Stats influence each Axie's performance during battle. Moves, strategy (the order in which moves are performed by each Axie), luck determine the amount of damage given and received by each participating Axie.