Aeon

The concept of  Æon stems from Thelema, a physical, mystical, and religious system created by the occultist Aleister Crowley in the 20th century. In the Thelemite religion, it is believed that the history of humanity can be divided into a series of those Aeons, each named after an Egyptian deity (the Egyptian gods were part of the Thelemite pantheon), and each was accompanied by its own forms of "magical and religious expression".

In the Toaru Majutsu no Index universe, the Aeons are a mysterious concept introduced by Aiwass when it meets Accelerator during the DRAGON Arc, when it states that Accelerator can't defeat it due to his attack being from the Aeon of Osiris. Aiwass itself is a being from the Aeon of Horus.

Aeons
The first of these was the Aeon of Isis, which Thelemites believed occurred during prehistory and which saw mankind worshipping a Great Goddess, symbolised by the ancient Egyptian deity Isis. It was characterized by pagan worship of the Mother and Nature.

In Thelemite beliefs, the Aeon of Isis was followed by the Aeon of Osiris, a period that took place in the classical and medieval centuries, when humanity worshipped a singular male god, symbolised by the Egyptian god Osiris, and was therefore dominated by patriarchal values. Christianity and other Abrahamic religions are considered by the Thelemites as being the representation of this Aeon.

And finally the third aeon, the Aeon of Horus, which was controlled by the child god, symbolised by Horus. In this new aeon, Thelemites believe that humanity will enter a time of self-realization and self-actualization. Aleister later came to believe himself to be the prophet of a new age the Aeon of Horus

Other Aeons
Some Thelemites believe that the Aeon of Ma'at will eventually replace the present one.

In relation to the story
In the series, several characters are referenced as being from or possessing the Aeon of Osiris, which is defined (in a simplified manner) as having their worldview defined by the God of Christianity. However, even this definition is too narrow, Accelerator, a non-Christian character, was also described as being in the Aeon of Osiris. Therefore, characters in the Aeon of Osiris have their worldview limited not specifically by the God of Christianity, but rather by an existential belief in outside objective systems of good and evil or right and wrong that are imposed on the character from without. Accelerator may not be a Christian, but up to a certain point, he is controlled by his belief in the notions of "heroism" and "villainy" as they are defined from an outside standpoint.

DRAGON Arc
When Aiwass' self defense program activated in order to counter Accelerator's attack, it referred to his Awaken form as being from the Aeon of Osiris and thus could not match against something like it from the Aeon of Horus.

World War III Arc
The topic of Aeons is mentionned yet again when Aleister Crowley and Fiamma of the Right face off at the end of World War III. There, Aleister praises Fiamma for having made it thus far, but scolds him for staying within the borders of the Aeon of Osiris, implying that, had Fiamma accomplished his plan under the Aeon of Horus, he would have succeeded. He then offhandedly mentions that he has himself gone beyond that, that he is now is of the Aeon of Horus, rejecting the old laws and wanting to create a new world with new laws. It is currently unknown how much are the Aeons of importance to Aleister's Plan. Later when Ollerus and Silvia arrive to rescue Fiamma after being left for dead by Aleister they offer him protection in exchange for what he had heard so they could try to read between the lines of the Aeon of Osiris that Fiammia is of and to see Crowley's Aeon of Horus which comes after.

Also, since the Invisible Thing sealed inside Touma's right arm was able to defeat Fiamma even in his La Persona Superiore a Dio form, it is safe to assume that the Invisible Thing might go beyond the Aeon of Osiris, a fact that even Aleister seems to confirm.