The Rule of thirds is dividing the image into thirds, so that the object, or the main subject in the image sits on the lines. This will make your photo look more balanced, and natural, also allowing the viewer to look at the interception point of the lines, and look at the background, and back to the main subject.
The images above are examples of the rule of thirds. The image on the left does not follow this rule, where as the image on the right does. The image on the left does not look very balanced and natural. The main subject of the image, which is the tree, is also right in the middle of the image, which takes the viewers eye off of the background of the image making it look un-natural and crowded. However, the image on the right does follow this rule, and the main subject is sitting on one of the lines. The foreground is also sitting on the lines. This gives the image more of a balanced and natural look, as the viewer naturally looks at the subject sitting on the line but also looks around the image at the background, making it look more natural, and then back to the subject of the image.