The Axis was a complex in inter-dimensional space, sort of "nowhere". Like the spindle and spokes of a bicycle wheel, the Axis was the hub that held the structure of damaged or truncated realities, dead-ends in time, formed by changes in established history which caused the flow of subsequent events to spiral out of control, often catastrophically. The hub regulated and maintained the realities and ensured that the contamination didn't spread to the primary timeline, like cauterising the wound to make sure the infection didn't spread. The timelines could be manipulated and even changed, but they would eventually return to their presumably time looped states. (AUDIO: The Axis of Insanity)
Romana II described the Axis as a tree with each branch being a deviant of the primary timeline. While all time streams were the result of temporal experiments, most of them the results of the Time Lords. (AUDIO: Reborn) Peri described it as all their mistakes tucked away in a cupboard. Because of this, the Overseer, the immortal and omnipotent controller of the Axis, had little regard for them. The Axis depended on the Overseer's mind to exist, and the Axis itself helped to hold the structure of time itself. If there was not a mind to sustain the Axis, it would begin to collapse and the beings held within the timelines would become aware of their existence.
The environment was constantly in flux, with time being extremely relative and space being illusory due to soft focus walls with a glow and bizarre perspectives. The environment could change, as when Jarra To made a carnival appear, and there were doors throughout the Axis leading to each timeline, with the Overseer monitoring everything from the Oracle room. (AUDIO: The Axis of Insanity)
Romana II, Leela, Narvin and Irving Braxiatel travelled to the Axis to find a Gallifrey that had a future for them, (AUDIO: Reborn) searching multiple Gallifreys before settling on one. (AUDIO: Forever) Eventually, the Daleks invaded the Axis and conquered a myriad of Gallifreys. (AUDIO: Arbitration)