Negative Space is can be viewed as the space around the subject, rather than the subject itself.  A simple example might be to look at this register to allow heat in.  The metal is the positive space, the open area cut out around it is the negative space.

In figure drawing or painting, you will commonly hear the figure referred to as the positive space, and the background as the negative space.

Here we have shown Mona Lisa, the positive space, in black, and the negative space would be the grey areas around her.

In some cases, negative and positive space can be switched, thus producing two different images in one.

The use of negative space is important to the feeling of life an movement in your work.  It will open up many possibilities to add interest to your work, giving space to breathe, and move in new ways.

Greek vases are an excellent source of study for understanding the use of negative space and rhythm.