A department was a major division or subdivision within the structure of an organization. Aboard Federation starships, aboard ship duties were delegated to each department, by each individual department head. Each department contained members of respective Starfleet divisions.
During events aboard a starship that required the crew's full attention, the captain often requested that all departments be on full standby, and may later require them to submit reports following said events. (TOS: "Tomorrow is Yesterday", "The Deadly Years"; TNG: "Angel One", "Lower Decks", "Genesis", "All Good Things..."; VOY: "Parallax", "Night", "Dark Frontier")
According to Luther Sloan, Section 31 was "an autonomous department" that operated with the intent of protecting the Federation's best interests, meaning that they "don't submit reports or ask for approval for specific operations." (DS9: "Inquisition")
During Amanda Rogers' visit aboard the USS Enterprise-D in 2367, Doctor Beverly Crusher had arranged to have her work in all the major departments aboard the ship while she was there in order to help her get an idea of what field she was most interested in. (TNG: "True Q")