There is a group of us. Over the centuries the members changed but not the purpose.
~ Emma Frost to Jean Grey in Wolverine and the X-Men
The Hellfire Club is a centuries old organization with roots in Britain during the 18th Century. It's members were infamous rakes who used their connections within the club to further boost their social standing and to indulge their sexual excesses.
Over time the club has evolved to include high-flying business men and women, as well as the profitable descendants of royalty and other Hellfire Club members. Being a member of the Hellfire Club is an illustrious boost among elite society. However, there is an even more esoteric sect of individuals who run the Hellfire Club, known as the Inner Circle.
The Inner Circle
The Club is run by a small group of influential members who form the infamous Inner Circle. Named after chess pieces, members of the Inner Circle will take on the titles of the Black and White (and later also Red) Kings, Queens, Bishops, and Rooks. The final position of Lord Imperial is reserved for the most powerful member of the club. Additionally, prominent members have also had assistants that did not have official titles within the club.
The Inner Circle was not always mutant dominated, but few members in several generations were not mutants. The Inner Circle dominates the club and secretly influences the club's activities in order to achieve even more power and wealth.
In the film X-Men: First Class, the Hellfire Club are the main antagonists, and consists of Sebastian Shaw (the leader), Emma Frost, Riptide, and Azazel. They intended to bring about nuclear war between the United States and Russia. Emma Frost was captured by Charles Xavier and Erik Lensherr, and imprisoned by the CIA, but Angel Salvadore joined the Hellfire Club after they attacked the Man in Black's base. Eventually, Shaw's attempts at starting nuclear war were foiled, and Shaw was killed by Magneto. Following this, Riptide, Azazel, and Angel joined Magneto's Brotherhood, along with Mystique, and they broke Emma out of prison. Thus, the Hellfire Club was no more, as its leader was dead and its other members were now part of the Brotherhood.
However, the ex-Hellfire Club members all met their doom before X-Men: Days of Future Past, as they were experimented on and killed by Bolivar Trask (although a viral marketing suggests that Azazel and Angel were shot by Project Wideawake operatives sometime before the deaths of Emma and Riptide, though both were shown in Trask's autopsy files).
X-Men Animated Series
The Hellfire Club renamed the Inner Circle (for censorship reasons) appears in the series consisting of the Black King, White Queen, Harry Leland, Donald Pierce and Jason Wyngarde. The Inner Circle aimed to control Jean Grey and brainwash into believing herself to be the Black Queen of the Inner Circle, though they only managed to unleash the Dark Phoenix and were defeated by the X-Men. Leland was injured, the White Queen, Pierce and Shaw later escaped with Shaw vowing vengeance against Wyngarde and the X-Men. The Hellfire Club appeared in the Dark Phoenix Saga storyline.
Wolverine and the X-Men
The Hellfire Club, once again renamed the Inner Circle, appears towards the finale of Wolverine and the X-Men. The group consists of Sebastian Shaw, Harry Leland, Donald Pierce, Emma Frost, Selene and the Stepford Cuckoos. Emma Frost joined the Inner Circle believing her mission to capture Jean Grey and take the Phoenix Force from her to destroy it. Emma later discovered the Inner Circle's true plan was not to destroy the Phoenix but release it and control the Phoenix for their own purposes. The Phoenix was released and the ceiling fell on Leland and Pierce while Emma used her telepathy to subdue Shaw and Selene.