The Federation came to being in 2161, on the planet Babel in the wake of the Earth-Romulan wars by five races - Terrans, Alpha Centaurians, Vulcans, Tellarites, and Andorians - for their mutual benefit and protection. Individual rights and civil liberties are guaranteed by the Constitution of the United Federation of Planets, signed in 2161. The Federation has expanded rapidly since, and today is composed of some 150 planets, bound together in a voluntary union for trade, exploration, diplomacy, and mutual protection. The Articles of Federation were signed by the United Nations of Earth, Epsilon Indii, the Alpha Centuri Concordium of Planets, Vulcan and many more, to provide a common defense against the threat of attack from the Romulan Star Empire. This constitution signed at Babel provided for freedoms and liberties for all races, a representative democratic government with universal suffrage, and established Starfleet, a government agency to represent, defend the Federation and learn more about the universe in general.
The Federation is governed by the Federation Council, which is made up of representatives from each member planet. It is headquartered at San Francisco, Terra. Presiding over the Federation Council is an elected Federation Council President, who makes his office at Paris, Terra. Each member world is entitled to retain its own form of government, its own military, and its own intelligence agencies. The planet Vulcan, for example, is governed by a the Vulcan Council and a group of titled ministers, possesses a small interstellar navy, and has an internal intelligence agency known as the V'Shar. Law within in the Federation is regulated by the Uniform Code of Justice.
The Federation uses a non-monetary economic system of "credits" to govern trade. Interstellar trade is largely free and unregulated, apart from certain contraband things such as slaves, drugs, and weapons of mass destruction. Because of its freedom and diversity, the Federation is wealthy and prosperous.
Within the interstellar borders of the Federation there are many non-member worlds. Most of these are less-developed worlds, declared off-limits by the tenants of the Prime Directive. These planets are being left to develop as they will under the protection of a Federation whose existence they are unaware of. When a world has reached the appropriate phase of cultural development, it is contacted and strongly encouraged to join the Federation. Other non-member worlds are neutral or are allied to the Federation. An example of the latter is Kaferia, a highly advanced non-Federation world located well within Federation space, which nevertheless has maintained strong ties with the Federation for centuries.
There is no one Federation culture. Some races, such as the Vulcans, have a unified society, religion, and cultural base. Humans, however, are unified as a people, but still hold different cultural values. There are nearly 150 member-planets in the Federation, creating a mosaic of cultures and traditions, bound together by their common interests: peace, order, liberty, and the pursuit of knowledge.
The members of the United Federation of Planets are represented in its government. Government is divided into three domains: legislative, Federation Council; judiciary, consisting of Federal courts and an Interplanetary Supreme Court of Justice, and; executive, led by the President. The Federation Council drafts laws for the Federation. The President, with the assistance of his or her Cabinet, enforces those laws and represents the Federation in foreign affairs. The Judiciary judges and punishes those who've broken laws and treaties with foreign nations, on the assumption that a defendant is innocent until proven guilty. The Judiciary may also rule laws and actions of the Federation Council or the President unconstitutional, as per the Articles of Federation. The Federation Council sits in San Francisco, on planet Earth. The President's office is in the city of Paris, also on planet Earth.
While the Federation is of a peaceful nature, its history is tainted with conflict. Since its very creation, the Romulans have remained an everlasting threat. In 2218, first contact was made with a warrior nation, the Klingon Empire. Poor relations in their early days led to nearly half a century of violence, until the Organian Treaty of 2267, followed by the Khitomer Conference of 2293. In the 2350's, war broke out between the Federation and the Cardassian Union, which lasted till a cease-fire and eventually a treaty in 2366 -- but even then, hostilities continued for years.