A bureaucrat was a member of a bureaucracy, typically in a government.
In 2152, Sub-Commander T'Pol used her experience with Klingon bureaucrats to secure Captain Jonathan Archer's release from Rura Penthe "for the right price." (ENT: "Judgment")
In 2268, Captain James T. Kirk noted his displeasure with his secret diplomatic mission to the Tellun system by commenting, "Some desk-bound Starfleet bureaucrat cut these cloak-and-dagger orders." Later, Kirk informed Elaan that she was to be properly prepared for Troyius, "as ordered by councils, rulers, and bureaucrats." (TOS: "Elaan of Troyius")
When Kirk was later kidnapped on Gideon in the following year, the efforts of the USS Enterprise crew to recover him were frustrated by the Federation Bureau of Planetary Treaties and Starfleet, who each insisted that the issue fell under the other's jurisdiction. First Officer Spock observed, "Diplomats and bureaucrats may function differently, but they achieve exactly the same results." (TOS: "The Mark of Gideon")
In 2367, rogue Starfleet Captain Benjamin Maxwell argued that if he informed Starfleet of his suspicion that the Cardassians were preparing for war, he would have to wait six months for bureaucrats to make a decision. When Captain Jean-Luc Picard questioned Maxwell's motives in attacking Cardassian targets, Maxwell remarked, "It smells musty in here. Like a bureaucrat's office." (TNG: "The Wounded")
While visiting Deep Space 9 in 2369, Captain Picard was due to discuss water management with ten Bajoran bureaucrats. (TNG: "Birthright, Part I")
In 2372 Gul Dukat argued that Kira Nerys's talents were wasted on Deep Space 9, commenting "On Deep Space 9, you're nothing but a bureaucrat, an administrator. But if you come with me, you can be a soldier again." (DS9: "Return to Grace")
As Tekeny Ghemor lay dying in 2373, he undertook the Shri-tal, passing political secrets to Kira Nerys. Kira found the ritual exhausting, however, as Ghemor appeared to have information on every bureaucrat and politician on Cardassia Prime. (DS9: "Ties of Blood and Water")
In 2374, when the Omega Directive was activated aboard the USS Voyager following the detection of Omega molecules in the Delta Quadrant, Captain Kathryn Janeway planned a risky mission to neutralize them. Realizing that the plan was Janeway's idea, Commander Chakotay stated that he always thought that Starfleet was run by duty-crazed bureaucrats, but that he found it hard to believe that even they would order a captain to go on a suicide mission. (VOY: "The Omega Directive")
When three of Kasidy Yates's cargo runs were reassigned in 2375, she observed, "Bureaucrats. Who knows why they do anything?" In fact, Captain Benjamin Sisko had "pulled a few strings" so that Kassidy could play in a baseball game against Captain Solok and his crew. (DS9: "Take Me Out to the Holosuite")
Wendell Greer, an alias of Section 31 agent Luther Sloan, was described as a mid-level bureaucrat in the Federation Department of Cartography. (DS9: "Inter Arma Enim Silent Leges")
After Commander Hilliard refused to release information on the Changeling morphogenic virus to Doctor Julian Bashir, Miles O'Brien observed that it sounded like a typical bureaucrat flexing his muscles, but Bashir dismissed this as a motive. (DS9: "When It Rains...")