One definition of hyperspace I read usually involves extra dimensions where the laws of physics are different than in our dimensions, allowing ships to go faster than light. It's a concept invented by an SF author in the early 1900s and used by many others, including Asimov in his Foundation novels.
I'm considering calling mine extraspace, but it probably fits the definition of hyperspace too. Specifically, all objects that exist in spacetime also exist in my version of hyperspace, which is merely a fourth "physical" dimension of those objects. If you hit an enemy ship's fourth dimension with an energy beam fired into hyperspace, that foreign energy is expelled from hyperspace at the point of collision between the energy beam and the ship's fourth dimension.
I don't actually mention hyperspace when it comes to the operation of my stardrives. I simply call it teleportation and leave the "how" to the reader's imagination. But I do refer to hyperspace (or extraspace) for how the hypercannons work. The firing ship doesn't itself enter hyperspace, but the cannons are able to fire into it, resulting in an energy beam that bypasses the target ship's shields (which only exist in the first three dimensions) and strikes the hull of the ship.
Dune had a book dedicated to the period/characters who invented their drive mechanism. They went into some details, but I don't recall how much.