Government of the Twelve Colonies
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The amount of unity between the colonies prior to the holocaust is unclear. Gaius Baltar claims to have gotten his Gemenese lab assistant a visa to work on Caprica (Taking a Break From All Your Worries). It is unknown to what extent this visa program was required for travel and transfer of labor between the colonies. However, this indicates that even unified under a central government the colonies were still independent to a certain degree . Furthermore, each colony had its own laws concerning various areas of life.
Before the first Cylon War, the Twelve Colonies functioned as autonomous states independent of their sister colonies. Each planet had its own localized government, civil services, and armed forces. Caprica, for instance, had a Prime Minister as the head of its government as well as its own Defence Minister and judicial system. Inter-planetary wars, civil wars, and other conflicts were common in this time, as inter-colonial relations varied widely (Caprica).
Before the Cylon attack
The president at the time of the Cylon attack was President Richard Adar. The president is assisted by a cabinet, consisting of the heads of the various government departments. Under Adar, the the Secretary of Education was Laura Roslin. A member of the Ministry of Civil Defense, known only as "Jack", is also identified (Miniseries, Night 1).
In the event of the death or incapacitation of the president, the vice president and the cabinet are included in the line of succession. At the time she assumed the presidency, Roslin was 43rd in succession (Miniseries).
The Colonial legislature is bicameral. The upper house is the Quorum of Twelve, with each colony represented by a single delegate, each with one vote. When the vice presidency becomes vacant, the Quorum is responsible for both the nomination and confirmation of the successor to that office. The president of the Colonies is also the presiding officer of the Quorum, with the authority to break tie votes. Ceremonial dress for Quorum members includes a sash representing their colony.
The lower house is apparently larger, and may be elected on a proportional basis. This may be the body which Gaius Baltar refers to as the "People's Council" (Lay Down Your Burdens, Part II). The government maintains a separate judiciary and recognizes an unelected religious body in an advisory capacity.
Ex Officio Advisers
The peoples of the Twelve Colonies appear to have a single religion based on worship of the Lords of Kobol. As such, a religious homogeneity exists that makes a "separation of church and state" unnecessary in the strictest sense. Government officials appear to be able to choose to consult religious leaders to support decisions or gather advice and consensus. Elosha, a priest, was also a member of the Quorum of Twelve for a number of years.
Government of the Fleet
The President of the Twelve Colonies of Kobol is directly elected by the population. For the most part Laura Roslin has presided over the Fleet while it has been spacebound, whilst Gaius Baltar presided over the time on New Caprica. On a few occasions Tom Zarek and Lee Adama have held the presidency for very short periods before the staus quo was re-established.
First Roslin Administration
Laura Roslin, Secretary of Education under President Richard Adar, assumes the presidency after the deaths of Adar and the 42 other secretaries and ministers above her under the Case Orange contingency system. She apparently never forms an official cabinet of her own, but employs various advisers during the remaining nine months of Adar's unexpired term.
Dr. Gaius Baltar is chosen as her vice president in a special, and hotly contested (with Tom Zarek) Quorum election 49 days after the Cylon attack. The extent of his duties are unclear, and he still has considerable time to devote to scientific undertakings.
Baltar is prepared to assume the role of the president during Roslin's imminent death from cancer, but the scientist cum vice president manages to save the president's life. Despite the action, Roslin suspects Baltar is a Cylon collaborator, and Baltar chooses to become a candidate for president in the upcoming presidential election.
Roslin's Presidential Advisers
Gaius Baltar announces his candidacy against Roslin one month prior to the presidential elections, with Sagittaron delegate and former opponent Tom Zarek as his running mate. Baltar runs on a platform opposed to religious interference in government affairs, prompted by the Rya Kibby affair, and in favor of colonization of New Caprica. Although Roslin attempts to fix the election in her favor, he is installed as President after the voter fraud is uncovered by Lieutenant Gaeta.
Immediately after his inauguration, Baltar signs his first executive order ordering the Fleet to return to New Caprica. One year later, President Baltar appears unsympathetic to the resource problems plaguing the colony and public approval of his administration is quite low. At the same time, a fleet of Cylon ships finds the settlement on New Caprica when they detect a trace radiation signature. Copies of Cylon models Five, Six, and Eight meet with President Baltar and his cabinet, and with Galactica and Pegasus having jumped away he offers the unconditional surrender of humanity (Lay Down Your Burdens, Part I and Part II).
After four months of Cylon occupation, Baltar is still in office although his role is reduced to that of a figurehead (Occupation). Tom Zarek refuses to cooperate with the Cylons and is summarily imprisoned in the New Caprica Detention Center until his execution is ordered by the Cylon Occupation Authority and signed by Baltar at gunpoint (Exodus, Part I). Baltar is deposed after Admiral Adama organizes and executes a daring rescue mission of occupied New Caprica's inhabitants. During the exodus, Baltar chooses to leave with the Cylons (Exodus, Part II).
Baltar's Presidential Advisers
After the escape from New Caprica, Tom Zarek assumes the office of the president, succeeding the missing President Baltar. President Zarek brokers an agreement with Laura Roslin whereby he will nominate her to be his vice president (subject to the approval of the Quorum of Twelve), he will step down as president, and she will become president again. In return, Roslin would then appoint him as her vice president.
Zarek's only major action taken in his brief tenure as president of the Twelve Colonies is an executive order authorizing a secret six-person tribunal tasked to find, charge, try, and execute collaborators and traitors. Shortly before stepping down as president, Zarek discloses this to Adama and Roslin. Zarek abides by his agreement with Roslin and goes through with his plan allowing Laura Roslin to resume her role as president (Collaborators).
Second Roslin Administration
Laura Roslin's first act as president is to grant a general pardon to anyone who collaborated with the Cylons. She abides by her agreement with Tom Zarek and nominates him as her vice president (Collaborators).
Roslin's Presidential Advisers
With President Roslin's health rapidly failing she elects to abdicate the presidency to aid Admiral Adama in his high-risk assault on The Colony. Romo Lampkin, former public defender and adviser to interim President Lee Adama is chosen to be her successor. Although Roslin survives the mission, she does not return to an active role in the government. President Lampkin takes part in the debate and planning of the settlement of the newly found "Earth", but expresses his concerns over Lee Adama's suggestion that the Colonials and Cylons abandon their technology and live like the native humans already present on the planet. With the survivors diffusing across the planet and possessing no means to communicate and co-ordinate with one another, it is assumable that the Colonial government is officially dissolved, thus making Romo Lampkin the last President of the Twelve Colonies of Kobol. His term is probably among the shortest in Colonial history, spanning just hours, possibly a few days at the most (Daybreak, Part II).
The Quorum of Twelve
46 days after the Fall of the Twelve Colonies, a new Quorum of Twelve is elected by survivors of the Twelve Colonies. It first convenes on Cloud Nine on the eve of Colonial Day and remains in session for some time thereafter ("Fragged", "Final Cut"). Cloud Nine is destroyed in a nuclear attack by the Cylon Gina Inviere. It is unknown whether any of the Quorum delegates are on board at the time of the attack (Lay Down Your Burdens, Part II).
After the commencement of the Second Exodus, then-President Tom Zarek oversees the swearing in of the new Quorum of Twelve, which was likely disbanded by the Cylon Occupation Authority. This Quorum convenes on Colonial One and consists entirely of new delegates (The Ties That Bind).
Vice Presidency (Day 49)
The new Quorum's first order of business is the election of a vice president, a motion presented by Tom Zarek (Sagittaron) and seconded by Dr. Gaius Baltar (Caprica). Marshall Bagot (Virgon) nominates Zarek for the position, and is seconded by Sarah Porter (Gemenon). Roslin's supporters select Wallace Gray to compete with him, but Gray is later replaced with Dr. Baltar.
Zarek loses the election six votes to six, with President Roslin casting the tie-breaker. Based on on-screen evidence, Zarek had the support of the representatives from Aerilon, Aquaria, Gemenon, Picon and Virgon, and his own vote as representative of Sagittaron.
Presidency (c. Day 280)
The late President Adar's unexpired term, which Roslin fills under the provisions of Case Orange, expires roughly nine and a half months post-holocaust. Roslin did not initially expect to run, due to the rapid progression of her cancer, but after her cure (Epiphanies) she makes it clear that she would stand for re-election.
Tom Zarek, defeated in his earlier bid for the vice presidency, privately states his intention to campaign for the presidency early on (Colonial Day), but withdraws his bid after he realizes that he could not hope to overcome Roslin's popularity. Instead, he throws his backing behind Roslin's vice president, Gaius Baltar, who publicly breaks ranks with the Roslin administration following the Kibby affair (The Captain's Hand).
The election is conducted by popular vote across the entire Fleet, with votes being counted by military staff on Galactica, overseen by civilian officials. The election resulted in victory for Baltar and a strong mandate for his policy of settlement on New Caprica, despite a suppressed attempt at rigging the election by the outgoing administration.
Vice Presidency (c. Day 800)
Following Baltar's disappearance in the aftermath of the Cylon occupation of New Caprica, he is succeeded by his former vice president Tom Zarek. Zarek indicates his intent to nominate Laura Roslin for the vice presidency before stepping down himself and allowing her to succeed him. Although unseen in the episode "Collaborators", this sequence of events apparently takes place without incident.
Post-holocaust relationship with the military
Based of the cited material that is available, this a tree of the Colonial Government system.