A bureaucracy was a form of government in which governmental decisions were made by unelected officials.
In 2365, Jean-Luc Picard explained that the Mariposa colony had been lost in the Federation's bureaucracy. (TNG: "Up The Long Ladder")
In 2368, when Dr. Beverly Crusher refused her request to use a genetronic replicator on the grounds that Starfleet Medical had already had rejected her three times, Dr. Toby Russell accused Crusher of hiding "behind the rules of some bureaucracy." (TNG: "Ethics")
In 2370, Constable Odo denounced the Bajoran Provisional Government as "idiotic bureaucracy" following as its decision to recall Major Kira Nerys to Bajor. (DS9: "The Circle")