Subspace and hyperspace are basic sci-fi jargon for an alternate dimension of the space-time continuum outside our perceivable 4D experience, in which the implications of Einstein's Theories of Relativity (namely that nothing can travel faster than light, and that while travelling close to the speed of light, time passes more quickly around you than your perception of it) are weakened or no longer apply. It's sci-fi hand-waving that is what the writers need it to be in order to allow the human race (and others) to travel faster than light and so visit other inhabited star systems in a convenient timeline.
The use of it is not universal in sci-fi. Arthur C. Clarke kept his stories primarily within the solar system and in general accordance with Einstein's theories; travel between Saturn's moon of Titan and Earth was about a three-week trip using a thrust engine based on an artificial singularity in Imperial Earth (basically the engine was efficient enough that the ship could be under power the entire trip, accelerating for half the journey and decelerating the rest). In 2001 and 2010 the ion-engine-powered Discovery vessels used cryogenic sleep to make the trip to Saturn's moon Iapetus possible (it was a journey of several months). The Host, loosely definable as sci-fi, talks of spaceships and interstellar sublight travel; travel between worlds colonized by the "souls" takes dozens, often hundreds, of years, mitigated by cryogenic sleep and the fact that souls have no natural life span of their own. Contrasting that, H. Beam Piper used hyperspace in his novels, but travel between Earth and most other colonized worlds was a few months' travel one-way. Lucas and Roddenberry obviously bought into FTL travel heavily for their universes, and travel between even rather far-flung worlds was a matter of hours or days.
Some of the newer universes have tied in with the ideas of string theory, some flavors of which indicate that we live in a universe of more than 4 dimensions, and our perception of only 4 is similar to a projection of a 3D object onto the flat plane of a piece of paper; it's subject to distortion based on the other dimensions that cannot be represented as such in our everyday perception. It then follows that if matter or energy can be moved along these additional dimensions instead of in the 3 normal spatial dimensions, messages or even ships can move along those dimensions and take advantage of their distorting effect on the 3 spacial dimensions, exceeding light speed as perceived by someone traveling sublight.