Hyper-Space refers to the higher dimensions above the third dimensions in and every sense of dimensions.
It is sometimes used to refer to Subspace Travel.
Few truly understand Hyper-Space but what is known is that the third dimension is situated within it. More advance minds state that all realities are situated in a Hyper Space and that it the diving force between reality and non-existence. Regardless what is known is that Hyper Space has much more ambient energy than 'normal' space, to the point that any links to hyper space can potentially allow disastrous amounts of energy to flow into normal space.
This refers to two means of travel
Entering Hyper Space to achieve FTL Edit
One cannot simply enter hyper space and expect to survive, ships that skirt into higher dimensions temporarily have special shield to hold back the intense energy and even then they do not stay. Note that 'entering Hyper-Space' does not mean they have entered 'Hyper-Space' as the term refers to all higher dimensions and there multiple
Subspace Travel Edit
Some refer to Subspace as Hyper-Space and when they enter Subspace directly they refer to it as Hyper-Space Travel. For them Hyper-Space refers to any dimension not their native own.
Hyper-Space Link Edit
Known also as a Hyper-Space Bridge to some, it is a link to Hyper-Space. It is functionally an energy source as the much greater Ambient energy of Hyper-Space, trickles down to the mortal dimensions as a vast power. A stabilized form of this energy is Rarified Ethium and natural Energon. The energy is difficult to control especially by mere mortals, but some of the 'higher races' and higher dimensional beings can control the power safely through their various means.