Hyperspace is a theoretical extension to subspace. According to current theories, the universe is made up of 29 spatial dimensions, all but the first four of which are sightly curled up into a ball (roughly planck length in diameter). Subspace travel happens in the first four dimensions. Hyperspace travel, instead, would forably uncurle one of those higher dimensions, allowing for a more direct travel from one point to another.
All points in the 3 dimensional space we're familiar with map to a point in a higher dimension. However, in one of the collapsed spatial dimensions, the distance between points are fractions of fractions of the width of subatomic particles. This means that in order to travel from one end of the galaxy to the other, one simply has to shift an infinitesimal distance in a higher dimension. This should allow for near instantaneous travel from any point in the universe to any other.
For the sake of conversation, these curved higher dimensions are referred to as "hyperspace".
As it turns out, even with cheap graviton generation, the energy requirements to punch through hyperspace is astronomical, and the gravitational sheer is so immense very few ships would be able to survive the travel. It also seems that the energy requirements raise exponentially with the mass of the object, meaning there's a rather hard limit on how large something can be to even punch through.
Hyperspace is not simply a collapsed sphere of space, but is rather twisted back on itself in a sort of quantum knot, the shape of which is unpredictable. This means that for reliable hyperspace travel your current hyperspace coordinates and your destination coordinates must be known with absolute precision. But, because hyperspace has been shrunk to a quantum scale, simply reading your coordinates alters them. And because coordinates are so close together, even the smallest variation could send you to the other side of the universe.
It is possible for skilled Focus users to predict hyperspace coordinates (or, more accurately, will their ship to exit hyperspace at the correct coordinates). This is, currently, the only known way to navigate hyperspace, though since the use of Focus is both unknown in general, and limited to specific species, the practicality of hyperspace is limited.
The only known use of hyperspace was the Archigos Experiment. Hyperspace travel was achieved through the generation of an artificial black hole as a power source. The destination was random, as the experiment was just to enter hyperspace and then return, without traveling. The return of the vessel through hyperspace was only possible due to several Focus users working together to effect the outcome of the hyperspace exit.