Fai's introduction is in chapter two of the manga, in which he is shown sealing away King Ashura of Celes country at the bottom of a pool. He transforms his creation Chii (originally from Chobits) into a net-like shield, instructing her to alert him if King Ashura should awake. He then uses his magic to transport himself through dimensions, to the shop of the Space-Time Witch, Yuuko Ichihara.

Out of his travel companions, Fai’s motives for joining the group are the least clear. He was the only one to arrive at Yuuko's shop purposefully, by use of his own means. His wish was to never return to Celes country, but as the price for this would have been too great, he travels worlds with the group instead. He fears King Ashura’s awakening, believing King Ashura will chase him through different worlds. The basis for this fear is as yet unknown.

According to CLAMP, the correct pronunciation of his name would rhyme with "eye", so instead of spelling it like "Fay", as they intended, they thought that this would confuse readers. Thus, they stuck with the spelling of "Fai".

Spoiler:
"When Fai's past is finally revealed, it is shown that "Fai" is actually named Yuui, and his twin brother is the real Fai. In Valeria Country, though the birth of the second prince's son was highly anticipated, the birth of twins was considered to be a bad omen. Shortly after their birth, the second prince died, and their mother committed suicide out of guilt for giving birth to them. In addition, the country suffered many misfortunes, and the twins, when combined, bore powerful magic that rivaled that of the current king.

The king decided that making them unhappy was the only way to break the "curse", and since both refused to kill each other, they were locked up separately in a place where magic is ineffective and time does not flow: a tower residing in a deep valley; Fai was imprisoned atop the tower, and Yuui was confined at the base of the tower. Both were condemned to stay there "until the world's destruction."

Despite their unhappiness, the number of bodies dumped into the base of the tower began to grow. Yuui tried endlessly to climb up towards Fai, so they could escape, but he did not succeed. Eventually, the king became insane, and he executed every person in the country, save Yuui and Fai. He then jumped into the valley himself and committing suicide in front of a horrified Yuui, reminding him that their birth was the reason for the country's misfortune, and that they would now "pay for that sin" by being the only people alive in Valeria.

Eventually, Fei Wong Reed appeared from another world, and offered to free one of them. Fai chose to save Yuui, resulting in Fai being thrown from the tower to his death. Fei Wong then convinced Yuui that he is at fault of his brother's death and that Fai might be revived, allowing Fei Wong to use Yuui's guilt in order to turn him into his pawn. In order to bring his twin back to life, Yuui allowed Fei Wong to bestow two curses upon him, one of which was removed from Yuui's memory. Fei Wong then told Yuui that someone would free him from the valley, and that he would have to go on a journey, protecting "a princess" (Sakura), and killing any obstacles along the way. Fei Wong then tells Yuui that when he finds a magical feather, he should put it into Fai's body to prevent it from rotting, and that he should begin his journey when he finds a second feather, which he should give to Sakura.

Later, King Ashura came to Valeria from another world in order to retrieve Yuui and took him to Celes. When Ashura asked what Yuui's name was, Yuui said it was "Fai". Ashura later gave Fai the last name "Fluorite", the name of a precious talisman mineral, and he eventually earned the title 'D', given to the highest magician in Celes. And thus, Yuui became the protagonist "Fai D. Flourite". He found his two feathers while helping villagers during an avalanche, and decided to use one to create a person, Chii, in his mother's image, Chii.

Ashura eventually tutored Fai in magic and comforted him whenever he felt unable to overcome his dark past. When Fai discovered that he could only learn "magic that kills people", Ashura told him that his smile held the ability to heal others. In exchange for everything that he had done for him, Ashura asked Fai to promise that he would eliminate anyone who threatened Celes.

Due Fai's first curse, Ashura traced a magical seal on Fai's body in the form of a Phoenix tattoo, which suppressed his powers from growing any further, and though Fai protested, Ashura told him that Fai's first curse would break after it was fulfilled once, and he wanted to be the one that Fai would kill. Ashura then erased what he had told him from Fai's memory.

Several years later, a mysterious and deadly monster began to kill Celes's innocent civilians. Fai swore to defeat the beast, but found out that Ashura was the monster. Fai, at first, blamed himself for what happened to Ashura, thinking that "the curse of the twins" was what had caused Ashura to kill, but Ashura told him that his magic grows when he kills, and that he always knew that he would become a murderer, even though he was not sure why. Fai realized that Ashura had brought him under his care for the sole purpose of killing him when he reached this state. Unable to bring himself to kill Ashura, Fai cast a spell on him, putting him into a deep sleep. However, no magic spell was permanent, so the king would eventually awaken. In order to avoid having to face Ashura when he woke, Fai used his magic to teleport himself to the Dimensional Witch, where he began his journey.

(Source: Wikipedia)

In Horitsuba Gakuen, he is the chemistry teacher.