Arcadia is, in theory, an absolute monarchy. In practice, as with any 'absolute monarchy', the ruler's authority is limited in ways which are more powerful than formal laws, precisely because they can't be easily changed. Discworlders combine an instinctive desire to have aristocracies and monarchs with a built-in cynicism about such people and a tendency to do things their own way. Certain things are the job of government, and people will tend to obey when government does such things. When government gets into other areas, they are more likely to point and laugh than to do what they're told.
Arcadia is ruled by a triumvirate of monarchs: Jacqui, Mutsumi, and Thoric. Any measure requiring royal action typically has to be signed off by two of them before it is carried out. (This was instituted to prevent things like 'Kneel Before Zod Day' happening again.) Important measures (such as actual laws and the like) are typically signed on by all three of the monarchs. While each of the monarchs possesses quite broad executive authority, the Rule of Two has nipped many a potential disaster or fit of anger in the bud.
The work of government is done through a set of ministries headed by ministers appointed by the monarchs. The staff of said ministries is mostly permanent, so long as they don't screw up, while those at the top tend to change much more frequently. This has sometimes lead those in middle management to resist promotion...
The major ministries are:
The eight chief ministers are sometimes refered to as the Octagon. Each wears robes of one of the eight colors of the spectrum (The head of the Ministry of Magic wears Octarine Robes, of course). They meet weekly with the Triumvirate (or as many of the Triumvirate as can be assembled), to discuss policy and any crises.
Each ministry is divided into various Departments. Most of those are not worth detailing, but a few have enough interaction with the common folk to be worth detailing:
The Arcadian Rangers are the special agents of the Justice Department, tracking down the most hardened criminals. Some are brave heros. Some are nutcases. All are very tough and skillful.
AS&T: Arcadian Semaphore and Trump
This branch of the Ministry of Magic runs the semaphore system (aka 'the clacks') which connects the cities of the Green Circle and greatly facilitates their communications. It also provides trump communications and delivery services for fees, and serves as a sort of post-office. They fight a lot with the Trump department of the Ministry of Spending Money, which regulates Pay-Per-Trump and Trump Entertainment. This branch of the ministry brings in a LOT of revenue for the rest.
Normally, the executive work of the monarchy is done through the channels of the ministries, but the three monarchs all have the authority to issue declarations and writs which order specific action by some branch of the ministry immediately. This is usually not done because it's rude and tends to sow havoc in the chain of command, but sometimes there is no choice. (If this starts happening frequently in a ministry, it's usually a sign that the heads of it are likely to lose their office soon...)
Laws are the area of Arcadian government action most subject to the Rule of Two, as it requires the signature of two of the three monarchs to promulgate any law. (Laws monkeying around with the structure of government are known as 'fundamental law', and require all three).
The repeal of any law requires the unanimous consent of all three monarchs.
Arcadia has no legislature, though many local communities have village councils.
All three monarchs have the power to grant pardons and sitting together, the three of them act as the highest judicial authority. Together, they appoint nine circuit judges who wander around the countryside on regular circuits, sitting in judgement for any case which could incur the death penalty or over a year's imprisonment. Locally appointed (or in some communities, locally elected) judges deal with misdemeanor offenses.
There is also 'The Land Board', a panel of nine judges who work in teams of three to deal with all land-related lawsuits, of which there are far too many.
Succession is rather a tricky issue; while there is an officially designated heir to the throne, the current system is attuned to the capabilities and necessities of a married trio, one of whom is unlikely to die without taking Arcadia with her.
The chief way in which the communities of Arcadia bring their complaints to the King is in the form of petitions. Some petitions simply state problems, others ask for various solutions, some are pre-written laws they hope the Triumvirate will sign off on. Petitions are typically brought to the Ministry of Home Economics, which then schedules their presentation to the monarchs during the 1-3 hours a day scheduled for recieving petitions.
Petititons are also sometimes presented directly to ministries when they deal with affairs which don't require actual royal action.
Discworlders both want to have nobles and want to make fun of nobles. Noble titles are heritable, but do not necessarily convey any sort of authority, though certain privileges are inherent in the ranks. The nobility system is similar to that of Amber.
Nobles who act properly will get respect, nobles who do not will tend to be relentlessly mocked by everything that can talk when their head is turned.