Awakening

Awakening (覚醒) is a term used by the English-speaking fandom of the Toaru Majutsu no Index to refer to espers that developed angelic powers and characteristics, such as wings and a halo. Specifically, only Accelerator — the most powerful esper in Academy City, has been shown to be able to "Awaken" throughout the course of the story.

Terminology
The term most likely originates from the fan-translation of the 15th Index light novel, where the narrative states that Kakine 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 Kakine'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 where 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 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. Kihara Amata, meanwhile, believes that Accelerator (unable to control any forces explainable by physics, as his Electrode was unpowered at the time) changed his Personal Reality to control the vectors of something beyond regular physics, such as AIM Diffusion Fields.

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.

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.

Accelerator's second awakening was after his fight with Kakine Teitoku, where he lost himself to blood-lust after witnessing Yomikawa Aiho's torture at Kakine's hands. His third awakening was against an avatar of Aiwass, who revealed that the devilish wings were a phenomenon originating from the Aeon of Osiris. It was not until his fourth awakening at the hands of Misaka Worst during World War III was Accelerator able to retain his rational thought processes, as seen by his argument with Touma in Russia.

In his latest awakenings, Accelerator manifested pale platinum wings,  comparable to those of Aiwass - a being of the Aeon of Horus,  having attained such a status by crossing the Clonoth's equivalent of the Abyss with the assistance of Qliphah Puzzle 545. In this state, Accelerator was able to embed the Clonoth into the world, causing a Holistic effect that temporarily separated Coronzon from her physical body.

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.
 * There was once the misunderstanding among the fandom that Kakine Teitoku had Awakened, however, this was cleared up by the primary fan-translator upon assuming his role in the translation projects. See here.
 * Rensa 29, while making use of Accelerator's ability, was also shown to use the powers associated with Accelerator's 'awakening', deliberately getting pushed into a difficult situation to draw out the black wings said to come out when Accelerator is cornered and manifesting the white wings while determined to protect Yakumi Hisako, though she didn't manifest a halo.