This is the upper house governing body and leading authority of the Twelve Colonies, a coalition of 12 different ethnic and species but related in some way to each other. The council formed what was essentially a federation, elected chairpersons of each individual race from each separate colony taking office on the council.
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").