Battlestar Galactica (1978)
In the 1978 version of Battlestar Galactica, the Quorum of Twelve—referred to there as the "Council of Twelve"—was the civilian government mentioned occasionally on the show. Most day-to-day aspects of civilian life were handled by the quorum; however, they typically left military decisions to Adama and his crew. The Council has its own security force, but interestingly uses Colonial Warriors for security in their own meeting chamber. There is little love lost between Security and the Warriors.
Commander Adama, in addition to his role as commander of the Battlestar fleet, was a member of the original Quorum. Adama is presumably the representative from Caprica, though how he held that office while commanding a Battlestar is a matter not addressed in the series. Adama was one of two members that survived the destruction of the colonies. After the destruction, Adama presides over a new version of the Quorum.
In "Saga of a Star World", Count Baltar appears as another member of the original Quorum, betraying the colonies on the condition that his colony was to be subjugated instead of destroyed by the Cylons. In "Lost Planet of the Gods", Baltar points out that his seal is the same as Adama's and so he was able to enter the pyramid.
A pivotal character who later disappears is Sire Uri, who is appointed to the new Council during "Saga of a Star World".
All of the Council members seen in "Saga of a Star World" have been replaced by the time of "Baltar's Escape" wherein the entire Council is captured in the Galactica landing bay while greeting representatives of the Eastern Alliance.
Battlestar Galactica (2004)
In the episode "Colonial Day", the civilian government was reconstituted, as elections were held for new representatives from the twelve colonies: Caprica, Gemenon, Tauron, Aquaria, Canceron, Sagittaron, Virgon, Picon, Libris, Aerelon, Leonis, and Scorpia. Each Colony (the surviving humans from those colonies) elects one delegate who votes on their behalf. The President of the Twelve Colonies acts as the chairperson. As seen in the episode "Colonial Day" the first issues they were forced to deal with included education, distributing of medical and food supplies, and the election of a Vice President. The first meeting coincided with the 52nd Colonial Day which celebrates the signing of the Articles of Colonization; the founding of a unified government for the twelve colonies.
When President Laura Roslin was arrested and the fleet placed under martial law following a coup d'état by Commander William Adama in orbit above Kobol ("Kobol's Last Gleaming"), Colonel Saul Tigh tried to justify Roslin's arrest and the suspension of civilian government to the Quorum of Twelve by showing them what he thought would be Roslin in an incapacitated and delusional state ("Fragged"). However after an impassioned speech to the Quorum (after which it was clear their loyalties lay with her, and not the military) Roslin led a civilian revolt which eventually returned civilian rule to the fleet.
It appears that the President can make appointments to the Quorum when vacancies appear in its membership; Lee Adama was appointed the Caprican delegate, apparently after the death or resignation of his predecessor ("The Ties That Bind"). It also appears that the Quorum can select the Acting President in case of the incapacitation of the President; it selected Lee Adama to be Acting President after Roslin's disappearance aboard a Cylon rebel basestar ("Sine Qua Non").
All members of The Quorum, except for Lee (who only survived a murder attempt thanks to Starbuck), are murdered by Galactica Marines at Vice President Tom Zarek's command during the mutiny for not supporting his coup ("Blood on the Scales"). Afterward, Roslin and Lee decided not to reconstitute the Quorum of Twelve, since the Twelve Colonies no longer existed. Instead, a new council was derived with representatives from each vessel in the fleet ("No Exit").