Hyperspace is a parallel universe in another dimension from normal space. This parallel universe has a rough point-to-point correspondence to our own universe, but the points are much closer together somehow. Because of this, a ship which shifts over to hyperspace and then travels for a given distance in one direction, then shifts back to normal space, will return to normal space several orders of magnitude farther away in the same direction. This allows starships to travel between star systems in days or even hours rather than in many years, without breaking the speed of light.
However, the equipment to safely enter hyperspace is large, bulky, expensive, and consumes immense amounts of energy, so most ships do not have the ability to enter or exit hyperspace on their own, and must instead use a Hyperspace Jumpgate.
Ships equiped with a gravity drive can form a jump point to cross into hyperspace when and whereever they want, although only warships the size of a heavy cruiser or larger are big enough to carry the massive gravity drive system.
Interesting Trivia: Many spiritual creatures and even the Technocracy knew of hyperspace by various definitions, the most common name being the Deep Umbra, although no one ever realized that its laws of physics and geometries did not correspond to regular space until rather recently.