The idea is that above everything we mere commoners know about, there is a secret world government or supergroup pulling the strings, controlling everything. Like, Every. Single. Thing. If it sounds absurd, it is. As Moo pointed out, it has history and plenty of it, moreso from conspiracy theorists.

I'm reminded of a quote from one of the guys who meets (or who has met, I cant recall) as part of the Bilderberg Group, who have been called pretty much everything the Illuminati have been called, only these guys do actually exist but just sit around and chat: When asked about the conspiracy theories of the Bilderberg Group this guy replied along the lines of "If we were a secret government of the world we should be bloody ashamed of ourselves."

Illuminati (plural of Latin illuminatus, "enlightened") is a name that refers to several groups, both historical and modern, and both real and fictitious. Historically, it refers specifically to the Bavarian Illuminati, an Enlightenment-era secret society founded on May 1, 1776. In modern times it is also used to refer to a purported conspiratorial organization which acts as a shadowy "power behind the throne", allegedly controlling world affairs through present day governments and corporations, usually as a modern incarnation or continuation of the Bavarian Illuminati. In this context, Illuminati is often used in reference to a New World Order (NWO). Many conspiracy theorists believe the Illuminati are the masterminds behind events that will lead to the establishment of such a New World Order.

To sum this up? Nothing. Conspiracy theorists believe this group exists. It doesn't.