A democracy is a form of government, which literally means "rule by the people," by the means of achieving a consensus by votes.
The inhabitants of Platonius followed the teachings of Plato, with a few adaptations, however, their society was, according to Parmen, "the most democratic society conceivable." He further stated that "anyone can, at any moment, be or do anything he wishes, even to becoming ruler of Platonius if his mind is strong enough." (TOS: "Plato's Stepchildren")
The Federation itself was of a democratic body. During the first Klingon-Federation War, Commander Kor told Captain James T. Kirk that he felt that the people of the Federation were much like the Klingons. Kirk noted that unlike the Klingon Empire, the Federation was a democracy, however, Kor clarified that he was not referring to "minor ideological differences," but instead meant similar "as a species". (TOS: "Errand of Mercy")
However, unlike the Federation goverment itself, life aboard Starfleet starships is operated by rule of the chain of command.
When the USS Enterprise was cornered by a First Federation buoy, the panicked Lieutenant Dave Bailey exclaimed to Captain James Kirk that they had phaser weapons, and that he voted they "blast it." Kirk, however, felt otherwise, stating, "I'll keep that in mind, Mr. Bailey, when this becomes a democracy." (TOS: "The Corbomite Maneuver")
During Captain Kathryn Janeway's plan to form an alliance with the Kazon, Lieutenant Commander Chakotay voiced his opinion about how "a lot of the crew feel they haven't had the chance to voice their opinions" about the situation. Janeway, however, held to her decision, stating, "this isn't a democracy, Chakotay. I can't run this ship by consensus." (VOY: "Alliances")