This page contains information regarding Star Trek: Strange New Worlds, and thus may contain spoilers.
The Federation Council was the unicameral legislature of the United Federation of Planets. Seats on the Council were filled by representatives from the various Federation members. (TOS: "Amok Time"; DS9: "Rapture"; VOY: "In the Flesh")
In 2371, as part of a ruse to get Ensign Harry Kim to buy worthless Lobi crystals, Quark told Kim that he should tell his commanding officer, Kathryn Janeway, that the Federation Council would be getting an official query from him regarding Kim being warned about Ferengi. (VOY: "Caretaker")
In late 2372, Chancellor Gowron sent a message to the Council, telling them that the Klingon Empire would not give up Cardassian planets which the Empire had captured. He also told them to withdraw Federation forces from the Archanis sector and Archanis IV. (DS9: "Broken Link")
In 2374, the Council sent a message to Grand Nagus Zek that Ensign Nog was to deliver. Jake Sisko believed it might have been a proposal for an alliance between the Ferengi Alliance and the Federation. (DS9: "Valiant")
In 2376, Zar informed Varn that "Captain Janeway" – i.e. Dala – had influence with the Federation Council and could speed up Varn's application for Federation membership. (VOY: "Live Fast and Prosper")
In 2286, the Federation Council deliberated what the Federation's position would be regarding the issue of the premature formation of the Genesis Planet. To that end, Starfleet Commander Admiral Morrow ordered the crew of the USS Enterprise to withhold their knowledge of the planet to anyone until they could make official policy. The Council also ordered that no one aside from the science team aboard the USS Grissom would be allowed to approach the planet. (Star Trek III: The Search for Spock) The Federation Council met three months later to consider the actions of Admiral James T. Kirk and his crew regarding his violation of these orders, their sabotage of the USS Excelsior and their theft and destruction of the Enterprise. However, in gratitude for the crew's work in saving Earth from the Whale Probe, the Council dismissed all but one charge and demoted Kirk to the rank of captain and, as befitting an officer of that rank, "sentencing" him to command a starship. (Star Trek IV: The Voyage Home)
In 2366, the Council was called into emergency session when a Romulan defector provided information on a possible Romulan invasion fleet arming near Nelvana III. They determined that the USS Enterprise-D, the Starfleet ship on site, should assume anything Admiral Alidar Jarok said was a lie. (TNG: "The Defector")
In 2369, the Federation Council promised Ambassador Ves Alkar safe passage back to his homeworld after bringing peace to Rekag-Seronia. Alkar expected Picard to fulfil that agreement, regardless of his, Alkar's, crimes. (TNG: "Man of the People")
Later that year, the Council imposed a warp 5 speed limit on all Federation vessels and vessels within UFP territory, except in dire circumstances. This law was revoked when improvements in warp drive were achieved. (TNG: "Force of Nature", "The Pegasus"; VOY: "Caretaker"; Star Trek: First Contact)
Shortly thereafter, the Council began debate on the ratification of the Federation-Cardassian Treaty. It rejected a request by a Human representative to re-negotiate the status of Dorvan V and ordered the evacuation of all Federation citizens from the planet. (TNG: "Journey's End")
In 2372, the Federation Council condemned the Klingon invasion of the Cardassian Union and agreed to provide twelve industrial replicators to help rebuild the devastated Cardassian industrial base. (DS9: "The Way of the Warrior", "For the Cause")
In 2375, Starfleet Admiral Dougherty. claimed that the Federation Council had been responsible for the decision to forcibly relocate the Ba'ku from their world in cooperation with the Son'a. After the USS Enterprise-E contacted the Federation, the Council announced they would perform a top-level review of the situation and halt the relocation of the Ba'ku. (Star Trek: Insurrection)
Later that year, the Federation Council was responsible for the decision not to give the Founders the cure to the morphogenic virus, as they felt it would give the Dominion the upper hand in the war. (DS9: "The Dogs of War")
In an alternate 2390, the Federation Council offered to drop the charges of conspiracy against Chakotay and Harry Kim if they returned a Borg temporal transmitter and stood down when those two tried to send a message back to 2375. (VOY: "Timeless")
- ENT: "In a Mirror, Darkly, Part II"
- LD: "Second Contact"
- Star Trek films
The exterior of the Federation Council meeting in Star Trek IV: The Voyage Home was described in the script as "Federation Headquarters", in the "San Francisco Bay Area."  Concept art for the shot, labeled Starfleet Headquarters, was produced by Ralph McQuarrie. 
One person who witnessed the on-set filming of the Federation Council in Star Trek IV: The Voyage Home was John Schuck, who portrayed the Klingon Ambassador therein. He recalled, "It was exciting, because around 4:00 a.m., all these other people who are in that assembly hall were there, so suddenly there were 25 makeup people, there were all these hair people, and you really had this throwback to the days of Hollywood extravagance [....] It was much more fun [than other movies], because in addition to real people, they had all these bizarre creatures that worked, that moved." (Starlog issue #138, p. 29)
In the novel Articles of the Federation by Keith R.A. DeCandido, the Federation Council is depicted as being housed in a fifteen-story building in Paris called the Palais de la Concorde (located at the real Place de la Concorde). The Council chambers are located on the first floor, while the presidential office is located on the top floor. In Star Trek IV: The Voyage Home, however, the Council is depicted as being located in San Francisco, and "In the Flesh" makes mention of the Federation Council building being replicated by Species 8472 in their recreation of the city. The historical archive screen in "In a Mirror, Darkly, Part II" also lists the council as being in San Francisco. However, it has never been explicitly confirmed in the canon that San Francisco is the primary or only meeting place of the Council, leading to some leeway for the novels.
In the earlier novel A Time for War, A Time for Peace, also by DeCandido, the Council itself determines who is qualified to run for president on the basis of anonymous petitions for candidacy. In Articles, it is divided into a number of "sub councils," the equivalent of a legislative committee, to which the president appoints members with the full Council's approval. The Security Council, for instance, deals with matters of Federation security, while the Judiciary Council can make rulings on judiciary matters (that seem to overrule those of the Federation Supreme Court). The novel also implies there is only one councilor per member, although "Rapture" makes mention of councilors needing to be selected for Bajor's admittance to the Federation; it was not clear if this meant that Bajor would have multiple councilors or if the character was simply speaking generally. In Star Trek IV, there are several Humans, Vulcans, Andorians, etc. seated in the Council chambers; however, any number of those individuals may be guests (such as Spock was).
In short, the make-up and location of the Council is somewhat vague and open to interpretation based upon canonical evidence. However, based solely upon on-screen evidence, the council appears to be a unicameral legislature.