So it's Halloween and you remember how awesome I am and decide to make me some pumpkin shaped sugar cookies (slathered with orange colored frosting of course). You roll out the dough, press the cookie cutter down, and remove the cookie. What do you have left? A pumpkin shaped hole in the dough. The whole is nothing but empty space, a lack of dough, it is negative space.
Now that I've completely confused you, take a look at the illustrations. In this example, the girl, the subject of the drawing, is the positive form. The rest, the background, the space around the girl, is the negative space. You may have a preconceived notion of what a girl looks like which probably contradicts what your eyes actually see. The negative space is just a shape; one you probably don't have any preconceived notions about. This forces, or allows, you to focus on the the shape itself as simply a shape, not a girl, not a pumpkin or a cookie, but just a simple shape. Sometimes it's easier to draw the negative space than it is to draw the outline of the subject.
As far as composition goes, negative space is every bit as important, and in some cases more important, as the positive form itself, but that's another topic altogether.