What is the method of classifying organisms based on their phylogeny?

A system of classifying organisms based on their phylogeny is called Binomial Nomenclature.

In binomial nomenclature, organisms are put into groups based on their body structure similarities to other organisms. Then those groups are broken down into smaller subgroups. Each organism gets a scientific name based on the two smallest groups it belongs to (genus & species).

Ex. An elephant's scientific name is Loxodonta Africana. Loxodonta is the genus group. Africana is the species group

Phylogeny: an organism's internal and external structural similarities to other organisms.

(Relating one organism to another based on their inside and outside body structures.)