Negative space is the area of a composition that doesn't contain the subject. You often see the definition of negative space given as "the space that surrounds the subject," but this isn't necessarily an accurate assessment. Usually, negative space surrounds the subject, but in some cases, the subject can surround the negative space, and in rarer cases, the negative space actually can become the subject.
So what makes negative space so important to an image? Well, negative space has a number of useful aspects when it comes to photographic composition. First of all, negative space can add a sense of balance to a composition by providing a counterweight to the subject. Negative space typically should have little intricate detail so that it not only provides balance to the image, but also can be effectively used to draw the viewer's attention to the main subject. By default, the subject becomes the most important part of the composition because therein lie the details that provide the point of interest in the composition. In product photography, a photographer often creates ample negative space within the composition to allow the designers a place to add copy without infringing on the actual subject matter.
Negative space creates a dynamic tension between the subject and the background by creating a point/counterpoint element that makes the image more dramatic, therefore attracting and holding the eye of the viewer.