The Colonial Government is a democratically elected government, headed by a President. The president at the time pf the Cylon attack was President Adar. Upon his death, he was succeded by President Laura Roslin, who had previously been the Secretary of Education. The President is assisted by a Cabinet. Each government department is headed by a Secretary. The Vice President and the Cabinet are included in the line of succession. (Mini-Series)
Each of the twelve colonies is represented by one elected member of the Quorum of Twelve. It is unknown if this is the only legislative body in the federal government. A second house would presumably be larger and have representation based on population. In the event of a tie during Quorum meetings, the President of the colonies casts the tie-breaking vote. At the current time, Gaius Baltar represents Caprica and Tom Zarek represents Sagitarron. The Vice President may be but is not required to be a member of the Quorum. (Colonial Day)