Q: What fills the empty space between galaxies?
A: The space between galaxies is "mostly" empty, but the distribution is far from uniform. I like to describe the overall structure as a "cosmic web".
First, one must recognize that "normal" matter (hydrogen, helium, and the other elements) does not dominate the gravity of the universe. Astronomers have abundant gravitational evidence that an unseen component, which we simply call "dark matter", surrounds all large galaxies and clusters of galaxies. The exact nature of this dark matter is as yet unknown, but particle physicists have proposed a number of candidates. Our best estimates are that the total dark matter in the universe is about five times more massive than all the normal matter. Hence, the dark matter dominates the gravity in the universe.
To understand how the dark matter is distributed in the universe, researchers run computer simulations and compare their results to the observed galaxy distribution [Side note: My Ph.D. thesis concerned supercomputer simulations of galaxy formation]. The most successful simulations show that the gravitational collapse of dark matter creates long, thin structures that we commonly call "filaments". The filaments join together at "nodes". The simulated galaxies generally form along the filaments and collect into galaxy clusters at the nodes.
The picture accompanying this blog post is a visualization snapshot from one such simulation. Purple represents the dark matter and white represents the galaxies. The pattern of purple dark matter filaments and nodes resembles an an irregular, three-dimensional spider's web - hence the nickname of the cosmic web. The white galaxies are found at the denser points along the web structure, with more galaxies at nodes, which are the densest points. The darker, relatively empty regions in between the filaments are called "voids". Because the filaments and nodes are collapsed, compact structures, the volume of the universe is mostly voids.
I wrote above that the space between galaxies is mostly empty because it depends on where one looks. Along the filaments and nodes of the cosmic web, there is some normal matter and dark matter, but at much lower density than in galaxies. In the voids, there is only extremely low density material.