Kingdom: Animalia (animals)
Phylum: Chordata (chordates)
Class: Mammalia (mammals)
Order: Carnivora (carnivores)
Family: Felidae (cats)
Genus: *Felis* (for domestic cats) or other genera for wild cats
Species: *Felis catus* (domestic cat)
So, to summarize, cats are animals, chordates, mammals, carnivores, belonging to the cat family. Domestic cats are specifically classified as *Felis catus*.