Awakening

Awakening (覚醒) is a term used by the English-speaking fandom of the Toaru Majutsu no Index to refer to espers, specifically, Accelerator and Kakine Teitoku—the two most powerful espers in Academy City, that developed angelic powers and characteristics, such as wings and a halo. As mentioned above, only Accelerator and Kakine Teitoku, the two most powerful espers in Academy City, have been shown to be able to "Awaken".

Terminology
The term most likely originates from the fan-translation of the 15th Index light novel, where the narrative states that Teitoku waved his "awakened" six wings to attack Accelerator.

There is apparently no evidence to support that "Awaken" is a special term designated by Kamachi, who has shown to be very meticulous in creating terminology, using both kanji and furigana to make unique and somewhat symbolic names for his creations. It is most likely that the "awakened" in the narrative refers to the wings becoming "truly active" in Teitoku's purpose for it, which is to attack Accelerator, who has attained Black Wings in their battle.

Nevertheless, the term somehow became spread through the forums and imageboards were the primary fandom of the Toaru Majutsu no Index franchise discussed the novels, eventually becoming an accepted term to refer to Accelerator gaining angelic properties.

The term "awaken" is never used in the light novels to refer to any action related to Accelerator or Teitoku regarding their wings ever again since the 15th light novel.

Principles
Index witnessed Accelerator's first incomplete awakening while he was fighting Kihara Amata, and she thought to herself that the power he used then was so great, that not even a Saint would have been able to handle it. She also claimed that while the nature of the power was similar to Telesma, it was also fundamentally different in some way.

The "awakening" process itself is actually quite simple yet not easily achieved. The esper must have an important revelation; a realization important enough to have an influence in the way the user lives his life. Since the usage of powers is related to the esper's Personal Reality, the awakening process might be described as similar to "maturing" or "growing up" mentally. Both Kakine and Accelerator have shown that based on the type of revelation, the power gained can be completely different.

Kakine Teitoku fashioned for his own, seraphim wings through his Dark Matter, but he did not awaken until the very end of his ordeal with Accelerator, when he witnessed his Black Wings for the first time. There, he began to understand the true nature of his power and had concluded that it could even defeat Accelerator's black wings. Of course, this all came to naught after he was torn asunder by Accelerator before his epiphany took ground. His revival later seems to imply that he has forgotten what he had learned, though ever was he curious of the wings that Accelerator had.

Accelerator's first awakening gave him huge black wings which he is able to utilize as blades. However, when Accelerator changes his way of thinking, his black devilish wings become completely white and he also gains a halo as well. The basis of Accelerator's awakening was revealed to be related to the determination he has when "fighting between life or death" as well of his own stance about "killing or not killing" his enemy.

Trivia

 * Index refers to her John's Pen Mode activation as Awakening, however she uses a different reading for the Kanji ( Awakening (覚醒める) ). John's Pen Mode itself also uses the term "Awakening" for it's activation. . It should be noted that the most natural writing for "mezameru" in that case is 目覚める (Mezameru), and thus 覚醒 (Kakusei) is forcibly used by the author. It should be noted that John's Pen Mode also displays wings.
 * So far, all "awakened" espers take on forms resembling angels, whether this is the case for all "awakenings" is unknown, due to those having the potential to "awaken" being so few.