The White Tower Schism is the name of the separation of the White Tower that would eventually lead to war between two opposing factions of Aes Sedai and later on to their eventual reunion, timely after the Battle of Tar Valon.
Elaida's White Tower
Under Elaida, Alviarin Freidhen was chosen as Keeper of the Chronicles but was deposed due to her disappearance during the Siege of Tar Valon. She was replaced by Tarna Feir. Of the Sitters newly raised, only the Reds and Shevan were not unusually young.
There was no Blue Ajah in the Loyalist Tower, and Elaida had the color removed from her stole and seat to show that she had dissolved the Ajah entirely.
Rebel Aes Sedai
The rebel Hall
The rebel Aes Sedai raised Egwene al'Vere as Amyrlin and Sheriam Bayanar to Keeper of the Chronicles. The rebel Hall of the Tower, called the Little Hall by the rebel Aes Sedai, was an inn before Salidar was abandoned years before. Each of the Sitters who had fled the Tower gained a place in this Hall, except one Blue who presumably died escaping. Except for Romanda Cassin and Moria Karentanis, the others were unusual choices for Sitters. See voting history for a full analysis of the rebel Hall's votes.
Unaligned sisters are the third faction created in the conflict, consisting of those that did not pick a side in this conflict. Among these are several of the sisters that weren't residing in the White Tower at the time of the instigating conflict. This faction is fairly large, accounting for a little more than a third of all Aes Sedai.
There are also some sisters who are of unknown affiliation in that they have usually only been mentioned before the events of the White Tower schism.
Elaida do Avriny a'Roihan led a faction of eleven Sitters that deposed the current Amyrlin Seat Siuan Sanche for her dealings with the Dragon Reborn. The other ten Sitters were not informed of the meeting at which they voted to depose her. The Blues were not informed pursuant to the rules for deposing an Amyrlin, but the other seven were likely excluded because they couldn't be counted on to vote as the former desired.
Of the ten, all but one, Seaine Herimon, left the Tower (although one of the Blues may have died during the struggle after Siuan's deposition was announced). The Blues and, it seems, Janya Frende genuinely fled the Tower. The other five were a different story.
Gawyn Trakand helped Siuan escape the tower. Several hundred sisters (around 330) fled the Tower as well, including the entire Blue Ajah and almost all the sitters who had not been involved in the deposition, save Seaine Herimon, who remained a loyalist. Min observes that most of the Green Ajah (traditional allies of the Blue) fled as well, while the other four Ajahs split nearly evenly. It is possible that some of the sisters who fled did not join the Rebels, but remained unaligned as they also did not wish to support Elaida.
Later it was revealed that five of those sitters had originally been sent by their Ajahs. The Heads of the five "unaligned" (i.e., not Blue or Red) Ajahs met and decided to send one Sitter per Ajah to help guide the rebels toward reconciliation:
In their places, the Heads directly appointed themselves or women younger than what was usual for a woman to hold a chair in the Hall. These replacements were only meant to serve for a short period—presumably until the five Sitters returned.
There were two other pre-split Sitters who were not invited to the vote to depose Siuan. Janya (Brown) left on her own out of disgust at the illegality; Seaine (White) was not instructed to leave, and decided to stay to keep the Tower whole. It is likely that Seaine was not sent so that each unaligned Ajah would have exactly representative in Salidar. This may be why Janya's replacement, Shevan, was not an unusual choice, as Janya may have been seen as a geunuine rebel.
In Salidar, the five Sitters found a rebellion led by Sheriam and five other women that was better organized than the Ajah Heads had anticipated. They divided into factions meant to keep the Hall paralyzed by playing Romanda Cassin and Lelaine Akashi against one another (Faiselle and Takima joined Lelaine, while Magla, Saroiya and Varilin joined Romanda). By the time the schism healed, Ajah Head Jesse Bilal believed some of the five Sitters had come to side with the rebels more often than the Tower. Indeed, many of them believed the rumors about the Reds setting up Logain as a false dragon, making it difficult for them to continue defusing the rebellion as their personal beliefs conflicted with it. However, the five were consistent in opposing Egwene's motions and pushing for negotiations with the Tower.
Aside from the five Sitters sent by the Ajah Heads, the Blues, and Janya, the rebel Sitters were too young or otherwise unusual. Among the Sitters chosen in Salidar, only Romanda and Moria were not really unusual. Romanda Cassin likely obtained her seat because of her previous experience (former Sitter and Yellow Ajah Head). The five Sitters seemed to have been responsible for choosing the Too Young Sitters from their Ajahs; Romanda thinks that Magla had insisted on Salita Toranes for the Yellow, though even Magla did not dare oppose Romanda's raising as a Sitter. However; this tactic may have backfired as the young Sitters generally voted as a bloc in favor of Egwene's proposals and other decisive action such as declaring war against Elaida. In fact, the five Sitters sent from Tar Valon generally found themselves virtually alone on the losing end of votes.
The White Tower was eventually healed when Seanchan forces attacked in the Battle of Tar Valon. Elaida was captured and the White Tower reunified under rebel Amyrlin Egwene al'Vere. The pre-split Sitters who fled the Tower all regained their seats in the Hall, in most cases displacing the Too Young Sitters or taking seats vacated by Black Ajah members. In fact, the Brown Ajah returned all three pre-split Sitters to the Hall upon reunification as none had been Black.