When you first begin meditating most people are overly concerned with their thoughts, sitting for any length of time, and worrying if they are "doing it right". However, after you develop your practice, you realize that meditation is as natural as sleeping or eating.
There are many types or ways to meditate with various different techniques employed to help calm the mind. However the end goal is always the same...to get into a state of non-thought, or rather to slow your thoughts down and just be. This is what is meant by the space between thoughts. It is awareness.
It is important because depending on your belief structure, meditation helps in realizing that we are not our thoughts, but rather a consciousness that sits behind the thought. Eckhart Tolle talks a lot about this in his book, "The Power of Now".
Great question and hope this helps. :)