The Quorum of Twelve is a fictional governing body of the Twelve Colonies of Kobol on the two television series. In both series, it was composed of one representative from each colony.
Series creator Glen A. Larson is a member of The Church of Jesus Christ of Latter-day Saints, one of whose governing bodies is the Quorum of the Twelve, one of many LDS references in the Battlestar Galactica series.
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 active members that survived the destruction of the colonies. (A retired third member, Sire Uri, also survived) 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 came out of retirement to be 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All of the Council members seen in "Saga of a Star World" have been replaced by the time of "Baltar's Escape" wherein the entire Council is captured in the Galactica landing bay while greeting representatives of the Eastern Alliance.