Anthropology was the science that explored the origins (evolution), physical development, cultural development (language, politics), biological characteristics, sociology, and beliefs (religion and philosophy) of a culture. By the 23rd century the field expanded to include the field of astral anthropology. (TOS: "The Galileo Seven")
One who studied the subject was generally known as an anthropologist. During the 23rd century, Starfleet vessels included an A&A officer, who specialized in archeology, anthropology, and ancient civilizations. (TOS: "Who Mourns for Adonais?")
A three people anthropology team, including the Doctors Palmer, Barron, and Mary Warren observed and studied the proto-Vulcan humanoid society on Mintaka III in 2366. (TNG: "Who Watches The Watchers")
Many personnel cards in the Star Trek Customizable Card Game have anthropology listed as a skill and many missions require it.