In science fiction, hyperspace refers to a region somehow alongside and connected with our own universe (in some cases displaced in an extra spacial dimension) which may be entered using some sort of energy field. While hyperspace is in some way anchored to the normal universe, it's much more compact, so traveling one mile in hyperspace is equivalent to traveling a thousand miles in normal space, or a billion miles, or a trillion. This makes for a very good explanation of faster than light (FTL) travel: pop your starship into hyperspace, move along for a while, pop back into normal space, and you end up light years from your starting point in original space, with only having spent several months, weeks, or even days in transit. In fact, hyperspace is the most common device used for explaining FTL in a science fiction story when FTL is a necessary for interstellar travel.
In many stories, a starship can not enter or leave hyperspace too close to a large concentration of mass, like a star; this means that hyperspace can only be used to get to the outside edge of a solar system, and then must use other means of propulsion to get to or from the planets closer to the star. Sometimes this is simply a plot device so that a starship can't easily escape by conveniently slipping into hyperspace, thus ensuring epic space battles.
Hyper-space is usually portrayed as either being completely empty, with a gray or black appearance, or as being full of shifting, luminous tendrils and clouds of... something; imagine a universe crammed full of nebulas.