The Rakata invasion
The split of the Je’daii into Light and Dark is referred to as the Force Wars.:
The planet Tython, in the Deep Core, provided the first-known conflict
between the Force’s light and dark sides. Disciples of the light side
(the Ashla) fought the followers of the dark side (the Bogan), in what
is now known as the Force Wars.
Star Wars: The Essential Atlas
They seem to have begun as a conflict between the Je’daii and the Rakata, and later become a conflict within the Order itself.
The Rakata were an enormous empire that drew their power from the Dark Side of the Force. They sought to conquer Tython in order to renew their connection to the Force, and to gain access to the Infinity gate thereon, a source of cheap travel1:
According to an interview with John Ostrander, the author of the Dawn of the Jedi series, this Rakata invasion is what led to the division of the Je’daii:
The second, and the event that begins to polarize dark and light
within the Jedi Order, will be set into motion when another species
enters the Tython system shortly before the Force Wars begin.
This makes a great deal of sense. The war between the Je’daii and the Rakata brought unprecedented violence, on a world that was previously in balance. Further, since the Rakata drew on the Dark Side alone, their invasion might have turned many in the Order away from the Dark Side as a source of power.
On the other hand, it might also have convinced some Je’daii that the Dark Side was the only viable source of power against the Rakata, or that only by using the Dark Side could they be as powerful as the Rakata were.
In particular, the use of forcesabers seems to have helped create a rift in the order.
Forcesabers were Rakata weapons that channeled the power of the Dark Side. The Je’daii adopted them out of necessity, in order to greatly improve their ability to fight the Rakatan Infinite Empire. Some within the order embraced this weapon, whereas others feared it—an early harbinger of the division between Jedi and Sith (though the latter name would not arise for a long time):
In order to use the weapon, one had to move dangerously close to being out of balance, allowing the dark side more control. This is evidenced, for example, by Xesh suggesting that it would be hard for a Je’daii to draw on enough negative emotions to use one:
The seeds were definitely laid for a split between light and dark in the Order.
So basically, the Rakata invasion disrupted the fragile balance between light and dark on Tython. We know what prompted the split, but we don’t know the details of the actual conflict. In any case, the darksiders seem to have lost, whereas the light-siders carried on the name Je’daii (more or less). They then left and refounded the Order (now the Jedi Order):
The more proactive Ashla warriors left Tython for good, accompanying
the offworlders home along a hyperroute soon erased by the Deep Core’s
churning stars. These crusaders became the Jedi Knights, and established their headquarters on Ossus (at that time the most distant world known to the Republic).
Star Wars: The Essential Atlas
The current Jedi Order is light side because it descends from the Ashla devotees who left Tython. The darksiders seem to have lost, and the adherents of the light side who stayed on Tython did not, it seems, carry on their traditions.
1: Dawn of the Jedi: Force War (Volume 1, Issue 1)