Zoology was a branch of biology which involved the study of animals and similar non-sentient lifeforms, typically conducted by a zoologist.
In order to repopulate the surface of Talos IV, in 2254, the Talosians offered their zoological gardens to Christopher Pike, so to help him reclaim the planet's surface with a variety of plant life. (TOS: "The Cage", "The Menagerie, Part II")
While considering the ramifications of hunting down and killing the Mother Horta, in 2267, which Spock deemed as a potential "crime against science," Captain James T. Kirk reminded him that their "mission is to protect this colony, and to get the pergium moving again. This is not a zoological expedition." (TOS: "The Devil in the Dark")
Jadzia Dax had an academic degree, with Premier Distinction, in the subject of zoology, among other fields. She earned it prior to being joined with the Dax symbiont. (DS9: "Dax")
A scene cut from "True Q" would have established that Amanda Rogers had been considering a career in zoology. However, this was just one in a series of extremely fleeting ambitions.