All vertebrate animals, that is, all animals having a backbone, belong in the phylum Chordata. Because reptiles are vertebrates, they are chordates.
Phyla are also divided up into many smaller groups, called classes.
Within class Reptila exist smaller groups called orders. Lizards and snakes belong in the order Squamata. Iguanas, of course, belong in suborder Sauria. Within Sauria are several families, including the family Iguanidae, in which green iguanas belong. These lizards are commonly called rock iguanas.