There are millions of animal species, and many still haven't been discovered.
Some of the basic types of animals are:
mammals, birds, amphibians, invertebrates, fish, and reptiles. and those get divided into smaller types, and so on and so forth, and you eventually reach the genus, then the species.
For example, you start with the Animal Kingdom, then you make your way down to the Class Aves, the Order Strigiformes, the Family Tytonidae, Genus Tyto, and finally the species Tyto Alba, A.K.A the Barn Owl!