The obvious answer to this is nothing, emptiness and a large void. The truth is not that simple. In the immediate vicinity of a star is its solar system. Each star will have its own solar system with its own set of planets, asteroids and possibly a kuiper belt. For purposes of this question, the solar system ends at the Heliopause, the point at which the star's solar winds meet the winds from other stars. The area between a star and the heliopause is known as the Heliosphere which our planet, Earth resides in. Other places will say the end of the of the Solar System is past the Oort Cloud which may or may not exist.
The Voyager 1 space probe broke through the Heliopause in August 2012, the exact date is debatable so it is now in Interstellar Space travelling towards Gliese 445 in the constellation of Camelopardalis. It should encounter the Oort Cloud in 300 years time. Unfortunately, it'll have stopped transmitting back to Earth by then. Hopefully, we'll have Interstellar travelling by then and overtaken it.
The space between stars will include radiation from other stars, dust, asteroids, comets and any far travelling space probes. The distance between these items will be immense. You would have to travel large distances in order to see any.
Between the stars will be rogue planets, asteroids and comets to name some. Rogue Planets for example are those that have broken free or been ejected from their solar systems. They'll also be brown dwarves, failed stars because they don't have enough mass to start nuclear reaction. These stars are dark and not very easy to spot. Dwarf Star
In addition to those above, they'll also be nebulas, supernova remnants and other deep space objects. What you won't find between stars are quasars as they are galaxies in their own right. A Quasar is an active galactic nucleus which is a supermassive black hole that spews out streams of gamma rays.
The difference in the space between stars and the space between galaxies (Intergalactic Space) is not much except you'd also have Intergalactic Rogue Stars, Rogue stars are stars that have escaped the gravity of the galaxy that it used to be a part of. It won't be just stars that are between galaxies, they'll be rogue planets, planets that no longer orbit planets. In addition to stars and planets, there will be the usual suspects again, asteroids, comets, dust etc.
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