Hyperspace is a mysterious dimensional plane that allows humans (and other races) to travel with faster than light speed, being the one and only thing that makes creating interstellar empires possible. The word 'mysterious' isn't used without a reason, since even despite using it for hundreds of years, there exist not a single theory that explains how exactly it is working. With Hyperspace Drive not fitting into any scientific theory to ever exist.
What remains, are speculations and conspiracy theories. Further fulled by extreme rate of suicides and insanity among hyperspatial physicists. Which can be considered a sort of tradition of hyperspace, dating back to mankind's first flight in hyperspace, with all people on a USS Discovery's bridge returning slightly... different.
Travelling through Hyperspace
Even describing hyperspace is a problem, since every person to travel through it sees it slightly different. Sometimes it a feast of ever shifting colors, sometimes a singular color spread to every point in space, and sometimes completely darkness, with not even a tiniest ammount of light visible in many light years of space.
The biggest problem of trying to observe it, is the fact that it isn't exactly healthy for mind. Staring into Hyperspace for too long might cause inrreperable mental disorders. Which is why all sentient species attempt to avoid being exposed to raw hyperspace (non-raw being one processed by sensors and computers), with human prefering modification to the visual sensors of their space suit to filter everything with potential negative effects on their sanity. This method is, however, still imperfect, capable only of minimalizing the chance to catch some incurable mental disorders, that's why using it is a last resort.
That was the biggest problem of seeing now... and now the biggest problem of traveling through it. Hyperspace is spatially unstable. Which allows to use it as a FTL drive, while also granting massive and nearly crippling disadvantages, especially debilitating during first generation of hyperspace drives. Distance from Solar System to Alfa Centauri is more or less the same in Realspace (with all changes easy to predict). In hyperspace it's different. The flight between two systems might take for example from three to seven days, simply due - as it was called - spatial leeway. Early hyperspace travel required exiting it multiple times to determine your position in Realspace, before attempting to reach the target once again.
This flaw was partially fixed with second generation hyperspace drives. A transmitters capable of transmitting signals through hyperspace for much greater distance were created, that could also transmit signals from Realspace into Hyperspace. It allowed setting hyperspace routes from system to system by setting a chain of transmitters in Realspace between them. It allowed much faster and safer transit, while avoiding stationary threats by creating routes around it. All of this while still being able to travel through uncharted lands, just with all the risks and hardships of the first generation hyperdrives remaining.
Second biggest problem of hyperspace travel is the need to reach the very edge of star's gravitational influence. For Solar System it requires flying all the way to Oort Cloud... and a bit farther. The only exceptions are points of gravitational disturbance like hyperspace planet, and even there entering or exiting hyperspace isn't too comfortable. What's more a ship also has to deccelerate to enter hyperspace, which makes all interstellar travels really slow and costly.
Hyperspace level of spatial compression varies. There exists Hyperspace Anomalies, an areas of extremely high level of compression. Making all sorts of sensors even more faulty. You also can't leave hyperspace there without dying. What's more, objects placed there and incapable of movement will slowly (really sloooowly) drift towards the center of the anomaly. Reaching it can actually take tens of thousands of years. It can be somehow described as hyperspace version of black holes. From certain point onwards objects are pulled faster that they could flee, to return and described what they saw.
Anomalies possess a certain quirk of being filled with archeotechs. Ancient, ruined and long abandoned ships of forgotten species, pulled by hyperspace and filled with technology worth untold billions.
One also has to mention hyperlane's. Hyperspace also possesses 'layers' of sorts (or at least that's how physicists attempts to explain this), and there are places where you can enter a higher one. They are two-way tunnels that allow extremely fast travel between two points in Realspace, of extreme, in fact nearly astronomical level of compression. That allow to fly through many thousand light years within minutes.
As a side effect, they can't be left midway. Third generation of hyperdrives allowed creation of artificial hyperlane's (extremely costly and requiring both sides to reach certain agreements and 'drill' from both sides). Fourth generation of hyperdrives allowed to detect and travel very weak hyperlanes, paving the way for establishing subsectors.