The Illuminati is a group that practices a form of faith known as "enlightenment". It is Luciferian, and they teach their followers that their roots go back to the ancient mystery religions of Babylon, Egypt, and Celtic druidism. They have taken what they consider the "best" of each, the foundational practices, and joined them together into a strongly occult discipline. Many groups at the local level worship ancient deities such as "El", "Baal", and "Ashtarte", as well as "Isis and Osiris" and "Set".
Adam Weishaupt, a professor of Canon Law at Ingolstadt University and former Jesuit, formed and founded a secret society called the Order of the Illuminati on May 1, 1776, in Bavaria, within the existing Masonic lodges of Germany. Since Masonry is itself a secret society, the Illuminati was a secret society within a secret society, a mystery inside a mystery.
NOTE: Look for Anthony J. Hilder, Jordan Maxwell, or Fritz Springmeier for more in depth study of the subject.