The Trinarchs are a pair of Artificial Consciousnesses created by the Ishvana. Paimon looks like a 3 meter-diameter metallic ball with manipulator tentacles coming off it. Marax looks like a three-headed dragon out of children's stories, while Bael resembles a huge technological centipede.
When the Ishvana decided to destroy all the ACs it had created, the Binarchs were two of the five survivors - the other three are the Trinarchs.
As the Binarchs and Trinarchs fled the Ishvana, these five Artificial Consciousnesses found a world - Lyad V in the Lyad system - that was rich in natural resources. They commenced mining them and producing highly-advanced AIs to serve them, planning to eventually conquer other systems for resources and expand their power. Like the Ishvana from who's mind they sprung, the ACs are single-minded and power-mad.
In time, the five split into two groups: one of two, and one of three. The former are called the Binarchs, and their servants Binaries. The latter are called the Trinarchs and their servants the Trinaries. The lines they split along were originally around whom to attack. The Trinarchs, desiring certain kith artifacts supposedly held by the Ibyssian Brotherhood, decided to attack them, believing themselves to be among the greatest kith users known.
The Trinarchs, over the objections of the Binarchs, attacked the Brotherhood in a lightning strike at the homeworlds of the Ixodon and its great spaceyards, hoping to cripple the Brotherhood's formidable navy. They are partly successful, wreaking enormous damage on the Ixodon's fleets, and are preparing to move on Otra IV, homeworld of the Corran, where they believe some of the artifacts were being held. They successfully weakened the planet's defenses and raided it, seizing two of the artifacts before the elders among the Lha Ti arrived, and began crushing the ships of the Trinarchs one by one with their immensely powerful void kith. The Trinarchs, having gotten at least some of what they wanted, retreated back to the Lyad system, believing themselves safe. They would quickly find out they were wrong.
The Ibyssian Brotherhood hit the Lyad system very very hard in retribution. The Binarchs and Trinarchs had to flee.
The Binarchs were extremely angry at the collective punishment brought on them by the actions of the Trinarchs and demanded the stolen artifacts in recompense. The Trinarchs refused, and as the two groups were fleeing the Brotherhood they erupted into battle with each other.
The Binarchs managed to take one of the kith artifacts, and they broke away from the Trinarchs, going their separate way.