The United Federation of Planets, usually just called "the Federation," is an interplanetary government in the universe of Star Trek. Its space navy, "Starfleet," provides the ships used by pretty much every protagonist in Star Trek, and its politics are the topic of a great many movies and TV episodes.
The Federation is founded in 2161, shortly after the Star Trek: Enterprise series but long before the Star Trek of the 1960's. Its original members are the planets of Earth, Vulcan, Andor, and Tellar: from there, the Federation gradually grows to incorporate more and more planets, including Betazed, Trill, Bolarus, and (by the end of Star Trek: Deep Space Nine) Bajor. The general MO for the Federation is to contact planets as soon as they develop the capability for interstellar travel, and then give the planet a formal invitation to join up. The Prime Directive bars the Federation from communicating with planets until they reach this technological watershed.
The Federation's immediate neighbors, with whom it enjoys varying periods of war, peace, and alliance, are the Klingon Empire, Romulan Star Empire, and Cardassian Union. From the late 24th century onwards, they also get to deal with The Dominion, an interplanetary empire on the other side of the galaxy connected to the Federation by the Bajoran Wormhole.
Although the Federation's history is fairly well-defined through the series, its politics are less so. Throughout the series and movies, there's a distinct impression that the Federation is basically run by humans, with the other three founding races occasionally pulling some power. This is probably most pronounced in the earlier series and movies: by the time DS9 and Voyager come around, the Federation seems a bit more pluralistic.
We know for sure that the Federation has a president based in Paris, and a Federation Council elected by planetary governments, also located in Paris. Its constitution, the Articles of Federation, was printed in The Star Fleet Technical Manual, although this is "unofficial" and therefore not "trustable" to many. The Federation's defense establishment, Starfleet Command, is based next to the Golden Gate Bridge in San Francisco.
While the Federation appears to have a currency-based economy in its early years, the advent of replicators in the era of the later series and movies leads the Federation into what's often described as "utopian socialism." Since there's no real shortage of anything, wealth becomes irrelevant, money disappears, and the Federation concentrates more on learning and leisure. However, the outer reaches of the Federation, which don't have replicators all over the place, still have strong capitalist economies, and a number of other interplanetary states continue to use money (most notably the Ferengi Alliance).
The UFP is often compared to the United Nations or the United States, although (in my opinion) the European Union is a much better real-world comparison.