There is a common analogy about the structure of an atom, such as the nucleus is a fly in the centre of a sports stadium and the electrons are tiny tiny gnats circling the stadium (tip of the hat to 'The Greatest Show on Earth') but what is in the space between the 'fly' and circling 'gnats'?
Short answer: The space between the nucleus and the electron is not empty space, it is filled with an electron cloud. (You will understand this answer better if you read the long answer)
Long answer: Firstly, physics is a description of what we can observe. Depending on the scale of which you are describing, physicists, over the years, have different mathematical descriptions describing at different scales.
When you ask about the space between the electron and the nucleus, confusion arises when trying to use a classical description to describe things on the atomic scale.
On the atomic scale, physicists have found that quantum mechanics describes things very well on that scale. And when we use quantum mechanics to describe particles, electrons, protons, neutrons, etc, they are no longer thought of as point like particles whizzing around like the description that you are confused about.
Particle locations in quantum mechanics are not at an exact position, they are described by a probability density function. The previous answers other people gave before me are links to pictures of a probability density function of Hydrogen at different energy levels in 3D space.
These probability density function's shapes are often known as "electron clouds". They show where it's likely to find the electron. The darker regions are places of higher probability of finding the electron and the lighter regions are lower probability of finding the electron.
An important thing to note is that the pictures show "white space". Its actually a smooth picture with the probability getting smaller and smaller so fast that those locations look "white". This means that the probability of finding the electron in those seemingly white spaces are extremely low.
Now what does it mean to "find and electron?" This is another question you will have to ask. Its not related to this question.
Maybe one should add to the analysis of @QEntanglement and the nice electron probability clouds in the illustrations in the other answers, that also the space between the nucleus and the electrons is teaming with the exchange of virtual particles between the electrons and the nucleus, necessary to create the potential which determines the energy levels of the particles. These will mostly be virtual photons with loops and stuff.
Edit. This post was made duplicate to a new question, but I think my answer there completes this answer of mine so I am adding the link.