As you look through your viewfinder, imagine there are lines dividing your image into thirds, both vertically and horizontally, thus dividing your image into nine equal parts (as shown in the above image). If you place important picture elements on these lines (or preferably on their intersection points) rather than in the middle of your image, you will end up with a more balanced and visually appealing design.
This rule works well for both vertical and horizontal pictures since the sections are equal in size.
Note, however, that while placing your subject(s) off-center has its visual advantages, there are certain situations when you will want to center the important details (or your subjects) within a photograph. Experiment with the Rule of Thirds next time you are taking photos, and see how utilizing this technique can add dimension and interest to your image.
Please feel free to comment on examples in which utilizing the rule of thirds worked well for you or didn't add a significant amount of visual stimulation within your picture. Also feel free to add any additional information that may contribute to utilizing this technique when photographing.