As a scientific concept, hyperspace is questionable and unproven. It is a vague term, describing the ability to shatter the speed of light by slipping outside of conventional four-dimensional reality, into an ether zone where you can travel from any point in the galaxy to another without being bound by physics. But as a literary device, hyperspace is indispensable: Without it, all science fiction would take place in our solar system.
Scifi writers have been trying to conquer the difficulty of moving their stories past the boundaries of our solar system since the Golden Age of the genre. The first known reference comes was in 1634, in Johannes Kepler's Somnium, which posited that demons could be enlisted to bring men to the moon, but it was Isaac Asimov who truly popularized the term and the concept in the minds of genre readers. Asimov's hyperspace -- described in countless stories from the 1940s to 1990s -- allows instant teleportation or "jumps" between solar systems with the aid of hyperspace drives. His protagonists invariably describe the feeling of a hyperspace jump as being one of "a momentary insideoutness."
Nearly every book series that takes place outside of our solar system features hyperspace. In Dune, computers are outlawed by religious decree, so hyperspace jumps are only capable by complex calculation and a mystical second sight provided by overdosing on the spice melange. In Hitchhiker's Guide to the Galaxy, Earth is destroyed to create a hyperspace bypass, and undertaking a jump is described as "being unpleasantly like being drunk."
But if anything, hyperspace is even more prominent in film.
Star Trek's warp drives are simply rechristened hyperspace engines, and probably the first scifi show or movie to slap some actual technobabble on space travel to plausibly explain its heroes physics-defying adventures through the cosmos. In Star Trek, warp drives don't take the ship through hyperspace, but using subspace: Layers of undefined reality existing below normal spacetime, which can be used to propel the ship at 1,000 times the speed of light (but not to teleport a ship instantaneously).
But Star Wars is the first and most prominent film series to use the actual term hyperspace. In the Star Wars universe, all safe hyperspace routes were scouted out and established 25,000 years BBY (Before the Battle of Yavin) by an unknown species. Navigators do not alienate from these safe routes, because of the risk of jumping into a black hole or planet. Hence, Solo's bragging about making the Kessel Run in twelve parsecs is actually a brag not about his ship's speed, but his own ingenious navigation -- he effectively scouted a new route.
The list doesn't end there, of course. Battlestar Galactica makes frequent jumps through hyperspace, and unlike Star Wars or Star Trek's bastardized takes on the concept, jumping between two points truly is instantaneous. The Stargate series of films and television shows also utilize an instantaneous form of hyperspace travel... though it only happens between the eponymous stargates of the show's title. Babylon 5 allowed FTL travel thanks to hyperspace jump gates. And even the Browncoat favorite series Firefly, which takes place in one huge solar system populated by dozens of terraformed worlds, found need of a hyperspace drive to explain the quickness of Malcolm Reynold's cross-'Verse shenanigans.
Without hyperspace, there'd be no space opera scifi.
