The Ordo Hydra is an extremist offshoot of the secret society known as the Illuminati that is dedicated to the use of certain extreme measures to destroy the existence of Chaos in the universe once and for all. It is composed mainly, but not entirely, of Illuminati who are also Inquisitors, some of whom occupy extremely high ranks in the Inquisition.
Its origins can be found in the pathological hate of Chaos by these Illuminati, coupled with their impatience over the uncertain outcome and very long-term timeline to fruition of the mainstream Illuminati organisation's plans. In their impatience, the Ordo Hydra's members used sorcery to conjure an entity from the Warp called the Hydra, which was a form of psychic "virus" supposedly controlled by the Ordo Hydra that was capable of infiltrating a human mind and completely dominating it.
Their plan was to infect as many human worlds in the Imperium as possible with this entity, and then through the use of this virus they planned to direct the uncountable masses of infected humans as a single hive mind to deliver a concentrated, collective psychic blow to Chaos within the Immaterium, which the Dark Gods would presumably be unable to resist. The Ordo Hydra believed that this blow could be struck against Chaos much sooner than the more long-term plans of the rest of the Illuminati would have allowed -- at most, in a matter of Terran centuries rather than several millennia.
The "mainstream" Illuminati were determined to stop the Ordo Hydra's plan since it would result in the de facto psychic enslavement of much of Mankind even if the plan had proved successful. There was also the real possibility that the Ordo Hydra's plan would have backfired, resulting in the creation of a fifth Chaos God within the Warp comprised of the collective psychic unconsciousness of the infected humans. Such an outcome would precipitate the complete domination of the galaxy by the Ruinous Powers and thus ensure the final victory of Chaos rather than defeating it.
The differences between the various Illuminati factions led to secret operations that involved the assassinations of rival Inquisitor-Illuminati, and the use of multiple double agents. In one instance, the "mainstream" Illuminati used a double agent named Zephro Carnelian to manipulate the Ordo Malleus Inquisitor Jaq Draco into joining the Ordo Hydra, hoping that Draco would realise the fearful implications of the secret Ordo's plans and act against it.
Draco was indeed appalled by the Ordo Hydra and its plans, and clandestinely traveled to Terra, where he became one of the few mortals to enter the Hall of the Golden Throne in the depths of the Imperial Palace. There he revealed his knowledge of the Ordo Hydra's plan to the Emperor of Mankind Himself. Imperial records do not currently indicate what measures, if any, the Emperor and the Imperium took against the Ordo Hydra after Draco's revelations.
- The Inquisition War (Omnibus Novel) by Ian Watson