Anarchy, literally a system without leaders, was the absence of political governance. An advocate of anarchy was called an anarchist. While social theories of anarchy held that it could result in societal harmony, the term was often used as a slang meaning disordered social chaos.
As a societal state
The Chalnoth homeworld existed in a state of anarchy, rejecting authority in all forms. (TNG: "Allegiance")
Before the intervention of John Gill, Ekosians were primitive and lived in a primitive state of anarchy. (TOS: "Patterns of Force")
As part of the Julian Bashir, Secret Agent holoprogram Anastasia Komananov asked Hippocrates Noah if his plot was motivated by an anarchist philosophy. Noah claimed he believed in an ordered world instead. (DS9: "Our Man Bashir")
As a breakdown of order
Anarchy was often thought of as the undesirable outcome of a breakdown in law and order. (TNG: "Frame of Mind")
In an alternate reality where Nazi Germany had invaded the United States, Winston Churchill framed the importance to continue fighting as by stating that "It must be world anarchy or world order". (ENT: "Storm Front")
In 2024, witnessing his partner's desire to ignore Benjamin Sisko and Julian Bashir because they were at the end of their shift, accused him of being an anarchist. (DS9: "Past Tense, Part I")
In 2268, after observing the state of society on Sigma Iotia II, spock deduced that it had to become united, or else descend into total anarchy. (TOS: "A Piece of the Action")
After Kirk suggested to the Providers of Triskelion that they free the Thralls, one of the providers argued that this would mean anarchy, and therefore starvation. Kirk then countered that the Thralls should form a self-governing culture. (TOS: "The Gamesters of Triskelion")
After the collapse of government on Tilonus IV, Jean-Luc Picard described the planet as being in a state of total anarchy. (TNG: "Frame of Mind")
In 2368 a Borg cube traveling through the Nekrit Expanse failed, and a number of survivors managed to transport to a nearby planet. They started regaining their memories, and old conflicts between the many species represented flared up again. The situation got out of hand, until those opposed to the conflict described it as anarchy. (VOY: "Unity")
Kai Winn Adami was concerned that Bajor would descend into anarchy and chaos if Shakaar Edon did not return the soil reclamators he had refused to give back to the Provisional Government in late 2371. (DS9: "Shakaar")
Similary, Bahrat lamented that if he left a murder unpunished, his station would descend in a state of anarchy within a day. (VOY: "Fair Trade")
The Q Continuum chose social order over anarchy. (VOY: "Death Wish")
Other uses
Vulcan administrator V'Las labeled T'Pau an anarchist in an attempt to discredit her. (ENT: "Kir'Shara")
One of Riva's chorus described himself as representing "the anarchy of lust" in Riva's person. (TNG: "Loud As A Whisper")