Here's a breakdown:
* Kingdom: Animalia (all animals)
* Phylum: Chordata (animals with a backbone)
* Class: Mammalia (animals with fur/hair, mammary glands, and give live birth)
* Order: Primates (mammals with grasping hands and feet, five fingers and toes, and relatively large brains)
* Family: Hominidae (great apes - gorillas, chimpanzees, bonobos, orangutans, and humans)
* Genus: Gorilla (includes two species: Western gorilla and Eastern gorilla)
* Species: Gorilla gorilla (specifically referring to the species of gorilla in question)
So, the scientific domain of a gorilla is Animalia (kingdom), but its classification goes much deeper.