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:
(Source: Wikipedia)
In Horitsuba Gakuen, he is the chemistry teacher.
