Can anyone Explain how The Federation works to me like I am five years old?

The Federation is a representative republic, with an elected president as the head of the entire interstellar state. An election is held every four years, and a president may serve for an unlimited number of terms.

Political and direct administrative power is held within the Federation Council, which is composed of one councillor from every Member World. There is no limit as to how many terms a person may serve as councillor. T'Latrek of Vulcan, for instance, served on the Federation Council for nearly a century. Each individual Member determines how its councillors will be determined; the First Minister of Bajor, for instance, nominates that world's councillor and the Chamber of Ministers ratifies him or her, while the electorates of many other Members elect their councillors directly.

The Federation government has several executive departments whose heads form the Presidential Cabinet, who advise the president on their issues of jurisdiction and run their departments on a day-to-day basis. Cabinet members can have strong influence on Federation policy based upon their work with the president and the appropriate members of the Federation Council.

By the late 23rd and 24th century, the capital city of the Federation is Paris, and the capital planet is Earth. The seat of government is the Palais de la Concorde.

• Memory Beta article on The Federation