Originally, this is the name of a secret society started by Adam Weishaupt in Bavaria in 1776. This secret society was later discovered and dismantled, accused of subversive activity. Ever since, allegations of this society or its direct successor having gone underground to form the secret core of the world's clandestine power structure have persisted.
Weishaupt was one of the earlier proponents of synarchy. He saw the right to rule as the prerogative of his society by virtue of its would be superior members, their superior moral character, their superior knowledge and understanding of what is good for humanity. The lluminati were to form a secret network penetrating all areas of life, from government to church to economy and military. The illuminati network itself was strictly hierarchical and compartmentalized, so that each member only knew his closest fellow conspirators.
Weishaupt was a former Jesuit, also a student of Gnosticism and various esoterica.
This type of structure most probably runs the world but it is improbable that this were Weishaupt's creation or idea.
In the present work, illuminati is generally used synonymously with 'consortium,' i.e. the higher levels of secret government, consisting of agencies and players not known to the public even by name.