PLEASE NOTE: This information is NOT CANON for the Elite: Dangerous setting.
The Intergalactic Naval Reserve Arm was a Federation and Empire cooperative. It's primary purpose was to develop technologies to defeat the Thargoids. Many pieces of FEU fiction incorporate elements of the INRA into their storylines.
War against the Thargoids
As the war against the Thargoids continued, it developed from a cutting-edge science organisation to the vanguard of humanity's struggle against the alien aggressors. Elite pilots were recruited from across human space and given the most advanced ships and technologies to help fend off the enemy.
In the end however, it was INRA's research roots that won the war: a Mycoid virus was developed and delivered to the Thargoid homeworld by a courageous INRA pilot. The virus destroyed the Thargoid's organic hyperdrives, halting them in their tracks and excluding them from hyperspace travel. The scattered pockets of Thargoid infestation were systematically eliminated.
With the end of the Thargoid War, INRA was officially disbanded, it's veterans spreading to the corners of the galaxy to continue to gather fame as elite pilots or to degenerate and live off telling tall tales in dingy bars.
After the war
All was not as it seemed however; INRA was still active, keeping an eye on human exploration of space. It knew that humanity should never explore out to where the Thargoids home world was, for fear of another war. Nor should humanity discover the despicable, virtually genocidal way that INRA had won the war.
Their silent vigil was without problem until the 3250's when Mic Turner and Meredith Argent, owners of the AAAI Shipyards in Alioth decided to find out exactly what happened to the Thargoids. Together, they designed and built a ship far beyond anything ever constructed in civilian shipyards. This ship had the ability to travel to the Thargoid homeworld. When the Turners Quest left human space, INRA hunted it down and destroyed it, killing Mic Turner. Undeterred, the Argent's Quest took off with an unknown pilot and managed to discover the alien homeworld and cure their diseased hyperdrives, allowing peaceful contact between the two species. INRA continues to watch events in human space, operating in the dark, always beyond the knowledge of both civilians and standard military.
Many stories are tied to INRA, that cannot ever be verified without being killed or joining their ranks. Such rumours include that they are based at the mythical planet of Raxxla.