(covers information from several alternate timelines)
The High Council (also referred to as the Imperial Command or High Command) was the legislature and ruling body of the Klingon Empire.
Located in the Klingon capital of Qo'noS, the First City, the High Council was comprised of some two dozen representatives of the most powerful houses and headed by a Council chancellor. The members of the Council were charged with overseeing the welfare of the Klingon Empire and its citizens, with each member usually heading a major department. When the Council was summoned to determine policies, the input from each department head was used. They met in the Great Hall.
The Council had a long history of political intrigue, as the Great Houses battled for control over the Empire. Assassinations, duels, and hostilities were common during power struggles. When the Council chancellor was deposed, either through assassination or other means, a Rite of Succession was performed to instate the new chancellor. As with all Klingon occasions, a dose of honorable battle was inevitable, after which the two remaining contenders were to fight it out for the coveted position.
Later that year, the Council gave General Martok a mission to find the IKS B'Moth, which had disappeared near the Cardassian border. They also told him not to not enter Cardassian space. However, the IKS Rotarran was required to do so, when the B'Moth was discovered to have drifted over the border. The Council did not reprimand Martok for crossing the border, as they viewed the rescue of thirty-five Klingons and the destruction of a Jem'Hadar fighter as justification for doing so. (DS9: "Soldiers of the Empire")
Later that year, Martok expressed doubts that the Council would accept him as chancellor, as he was a common man from the Ketha lowlands, in Ketha Province. Worf believed that Martok's reverence by the troops would force the Council to accept him. Worf later challenged Gowron, calling him unfit to lead the council. (DS9: "Tacking Into the Wind")
In an alternate timeline in which Captain Benjamin Sisko suffered temporal displacement, Worf was able to persuade the High Council to let the USS Defiant enter the Klingon-controlled Bajoran system to attempt a rescue. (DS9: "The Visitor")
High Council members
The term "Klingon High Command" is mentioned in "The Trouble with Tribbles", Star Trek V: The Final Frontier, Star Trek VI: The Undiscovered Country, "Sins of the Father", "Dramatis Personae", and "Apocalypse Rising".
The notion of a Klingon High Council was established in a Klingon-centric memo which Ronald D. Moore wrote to Michael Piller at the outset of working on "Sins of the Father". This memo stated, "A High Council is the ultimate authority in all matters of Klingon life [....] The High Council is formed of leaders of each family of importance in the Empire." ("Sins of the Father" audio commentary, TNG Season 3 Blu-ray)
In "Reunion", K'Ehleyr is offered a seat on the High Council; yet, in "Redemption" and subsequent episodes, it is said that women may not serve on the Council. Concerning the place of women, Ron Moore commented, "I co-wrote both 'Reunion' (in which K'Ehleyr was offered the Council seat) and 'The House of Quark' (in which Grilka was told she could not serve on the Council because she's a woman). The reason for the change was: a) to service a plot element in 'House of Quark'; and b) to differentiate the Klingons from the UFP and the Romulans. The idea was that the Klingons were a traditionally patriarchal society and that while many elements of that have disappeared over the years, the Council itself was still the province of male warriors. This is not an endorsement of that idea, but rather an attempt to make them different than us. For example, their government is not a democracy, but rather an oligarchy ruled by powerful Houses, with an Emperor as head of state and we certainly aren't promoting that either!" (AOL chat, 1997)