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 of different colors (black, white, or platinum) 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, though it may not be limited to him.

For ease of reading, Awakening will keep being used in the article as if it were a legitimate term.

Terminology
The term most likely originates from the fan-translation of the 15th Index light novel, where the narrative states that Kakine 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 Kazuma, 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
While the exact origins of the wings is unclear, some conditions for "awakening" can be gleaned from the instances they appeared. The manifestation of the wings are shown to be intimately linked to Accelerator's mental state, either him having an emotional breakdown or breakthrough. Kakine Teitoku theorizes that it requires Accelerator's emotions to be at an extreme high or at an extreme low, whether they are negative or positive. Rensa 29 theorizes that the black wings require for Accelerator to be truly cornered for them to appear, and as she is able to engineer such a situation and gets black wings from it, she appears to be right.

Accelerator's first awakening gave him huge, 100 meters long black wings, but when he later changes his way of thinking and "gained a certain something", his black devilish wings become completely white and he also gains a halo. Based on the situation, where he admits to caring about Last Order and wanting to protect her, as well Rensa successfully getting the same wings when she is determined to protect Yakumi Hisako (though she didn't manifest a halo), it seems implied that the condition for white wings is a desire to protect.

Another condition for "awakening" seems to be contact with magic, and incorporating it within one's Esper powers. When first manifesting a particular color of wings, it follows Accelerator having had contact with magic: redirecting the explosion resulting from Taowu's body collapsing into space; hearing Index's singing prayer to free Last Order from her coma induced by the manifestation of Fuse Kazakiri; singing a magic spell to save Last Order using data from Index's song, his encounter with Aiwass, and magical parchments under a Telesma-filled sky; and crossing the Abyss using the new kabbalistic tree Clonoth with the assistance of Qliphah Puzzle 545. The association with magic is further reinforced by Index, who, when she witnessed Accelerator's first incomplete awakening, compared the matter of the wings as similar to Telesma, but also fundamentally different in some way. Indeed, after first manifesting the wings, Accelerator is able to sense magic in the form of trembling fingers and a pressure in his chest, the exact same symptoms experienced by Sasha Kreutzev, a former vessel of Telesma. Accelerator also speaks the garbled language of the Angels, like Archangel Gabriel and Aiwass, in the early manifestations, though he can now retain his rationality. Aiwass also calls the black wings a product of the Aeon of Osiris, a concept in the magical discipline of Thelema, though it is unknown what Aeon the other colors fall under.

On the Science Side, Kihara Amata believed 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. Kakine Teitoku thinks similarly, saying the wings "leave the category of logic altogether" and "no theory can explain" this phenomenon.

The phenomenon of "awakening" may not be Accelerator's sole purview. While fighting to save her mother, possessed by the Rosicrucian Spiritual Item violin Stativarius Tam Lin, Sakibasu Yuri used her own Stativarius Ainsel in battle, with her esper power Carbon Search helping her eliminate the rejection that normally occurs when an esper uses magic. At the climax, she temporarily gained white wings and a halo, presumably because she was fighting to save her beloved mother.

The wings have their weaknesses. Accelerator cannot maintain them for long and does not have complete control over them.

Academy City Invasion Arc
Index witnessed Accelerator's first incomplete awakening, which gave him huge black wings, 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.

Battle Royale Arc
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.

DRAGON Arc
His third awakening was against an avatar of Aiwass, who revealed that the devilish wings were a phenomenon originating from the Aeon of Osiris.

World War III Arc
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.

Later, his black wings become completely white and he also gains a halo. Based on the situation, where he admits to caring about Last Order and wanting to protect her.

Coronzon Arc
In his latest awakening, 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.

Necromancer Arc
The Toaru Kagaku no Accelerator manga retcons the first apparition of the black wings to having been during the Taowu incident at Seiin High School.

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.
 * 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. Kakine later talking about the wings as distinct from his Dark Matter confirms this.