The Flame is an ancient symbol, though its original meaning has been forgotten by present-day people.
In the Age of Legends, the Flame, along with the Dragon's Fang, was part of the ancient symbol of the Aes Sedai. Since the Breaking and the splitting of the Aes Sedai to remove the men, the Flame has taken on another meaning. It is no longer connected with female channelers by the common person, and its origins are lost to common knowledge by the passing of years, but superstition still regards the Flame as a sign of female Aes Sedai.
A new weave
During the Last Battle Egwene al'Vere, while observing the indirect effects of balefire on the pattern, and specifically on a rock, experimented a new weave that was able to fill the void in the pattern with tiny crystals.
Later Egwene, while battling M'Hael, used again this weave as a direct counter and opposite to balefire, a weave of pure creation as opposed to balefire's pure destruction. She called it the "Flame of Tar Valon". This "Flame" used at the maximum power that the Vora sa'angreal could provide was able to stop a blow of balefire and then to crystallize the body of Mazrim Taim from inside out like freezing him in ice. A multihued, beautiful crystal grew from him, uncut and rough. Somehow Egwene knew that the Flame would have had much less effect on a person who had not given himself to the Shadow.
Just moments after stopping Taim, Egwene realized that she drew far more power than she was able to wield safely, consequently she released it all around herself using an explosion of the Tar Valon Flame weave, and so she died while hitting and saturating with the One Power everything around her on the Polov Heights.
As a consequence the entire area which was badly damaged by the balefire got covered like a blanket of crystals coating the ground, filling all cracks with crystal and healing them; also all the Sharans that were surrounding her were immediately frozen into crystals. Egwene herself turned into a giant pillar of crystal column, wide as an ancient leatherleaf tree, rising some fifty feet in the air having Vora's sa'angreal frozen in its center.
Brandon Sanderson has stated that other Aes Sedai witnessed Egwene's weaving of the Flame of Tar Valon; thus, although Egwene never directly taught it to anyone, this important weave was learned by supporters of the Light.