The Illuminati (Latin for the Illuminated) was a secret society whose master plan was to combat superstition, religious influence over daily life, and abuse of state power. While their ideas were noble, their members had a tendency to disagree with eachother, especially about adopting Freemason ideollogies. They ultimately were disbanded when the government of Bavaria banned all secret societies. Which just shows how good they were at getting their original plans accomplished.
Of course, this doesn't make for a very good story, and the Illuminati is the center of all sorts of conspiracy theories, especially those where the endgame is a New World Order. Theorists posit that the Illuminati is responsible for all sorts of events, from the French Revolution to the JFK Assassination to 9/11, all of which is done to make their evil agenda of controlling the world a reality! This positioning of the Illuminati as an all-evil cabal hellbent on world domination has made them a popular villain in all sorts of media, especially in cyberpunk and paranormal settings for tabletop games (e.e. the aptly-named Illuminopoly).
In Warhammer 40k, the Illuminati were a group who sought to round up the Emperor's descendants, kill them in front of the Golden Throne, and kill the Emperor, all to reincarnate him into the Star Child. This was dumb, though, and the Inquisition declared them to be a Tzeentchian cult and purged them all. The idea of the Illuminati was re-purposed into the Horus Heresy-era Cabal, which is much cooler and way better handled.
All of this mystique around the idea of the Illuminati led to a fantastic card game being made by Steve Jackson Games, wherein the players are all various organizations, each seeking to enact their agendas over the world.