How is taxonomy used to determine closely two organisms are related?

The order of taxonomy is (from most to least broad): Kingdom, Phylum, Class, Order, Family, Genus, Species. Many things can be classified under Kingdom, but only a few organisms remain as you file into a Species. For example, dogs are in the same kingdom and cats, wolves and basically all other animals. However, as you file into family, dogs are still under the same catagory as wolves, and as you get to species, only different breeds of dogs remain.

Organisms are grouped into taxa based on similarities in their morphology, genetics, behaviour, and other characteristics; the closer the organisms are in their taxonomic classification, the more closely related they are thought to be. Taxonomy is a hierarchical classification system based on shared characteristics that determines how closely related two organisms are.