The rule of thirds is a guiding principle in photography, used by many photographers to emphasize key elements in their pictures and to create tension. This creates a different feel to the end result than simply centering the subject would.
The basics of this technique are quite simple: divide your picture into thirds vertically and horizontally, creating 9 equal squares, as shown in the photo to the right. Notice how the trees and the horizon fall in relation to the lines dividing the picture. The main subject of the photograph isn't centered, but the composition brings more interest to the photograph.
Use the intersections of lines to place elements of particular interest.
This principle can be applied to any photographic scenario: people, landscapes, groups, etc. With people, place eyes, lips, or hands on or near the intersecting lines. With landscapes, place the horizon near one of the horizontal lines instead of in the center of the photo. With groups, use this to help you with arranging the group into a pleasant formation.
There are many advantages to using this principle, and as you use it you will find out new ways to make your photographs interesting.