Hyperspace is a parallel universe made out of some kind of ether/energy that is stacked on top of our universe, while the latter only consists of the time-space continuum. It’s not easy to explain how it really works, but let me put it this way:
Three Dimensional Hyperspace
The space we are all familiar with is three-dimensional. This means it’s possible to go forwards and backwards in one direction, up or down in another direction and left or right in yet another direction. That’s all there is to say about that. Hyperspace however has more dimensions than just these 3 directions. Just like our universe, hyperspace has X Y Z coordinates, although they are more complex since they have 6 dimensions instead of just three axes. In this hyperspace, there are 3 coordinates which they call the X Y Z axes. But for simplicity’s sake let’s just add a few more letters and numbers to our axes since these 3 don’t cover all of the 6 dimensions. We’ll add U V W and P where U stands for up, V is for forwarding in the direction of travel, W is sideways left/right and P would be perpendicular to your direction of travel. Imagine a rocket in space going forwards in the Z-axis at full speed while simultaneously going upwards in the W-axis at full speed as well. This means that even though it is going forwards, it also dips down really fast according to our perspective from outside which can be defined by the X-axis. The ship is basically wobbling up and down really fast.
The 3D space and Hyperspace
This entire 3D space with all of its coordinates floats on top of the hyperspace, which has 6 dimensions. Those are not easy to grasp for us humans who are stuck in our boring three-dimensional universe, but luckily for us, there are beings called “Shapers” that can alter their genetic structure making them able to sense these higher dimensional energies known as the hyperspace. Since they have always been researching this parallel universe, they knew how to create portals allowing them to travel between universes. They also changed themselves so that their bodies were able to be sustained by this new energy source known as the Xel’Naga energy. This energy is what defines everything in hyperspace. It fills the 6 dimensions of hyperspace just like how our time-space continuum is filled with matter, gravity or magnetism. In fact, this hyperspace energy is probably the most mysterious out of all these forces. We humans have been able to create magnetism by manipulating a magnetic field, but our attempts with the Xel’Naga energy have failed miserably so far…
Travel through Hyperspace
So how exactly can someone travel through hyperspace? Well, there are 2 ways: Either build a portal yourself which will simply open up a wormhole that takes you directly into hyperspace or steal one from another race known as “The Protoss”. The latter sounds way more convenient, right? So just take your good old Battlecruiser and fly it through an opponent’s base shooting at everything in sight while trying to avoid/kill his army until he has no work left to warp back home.