The rule of thirds is one of the most basic principles of photography and an understanding of it can greatly improve the quality and composition of your photographs.
The human eye has natural tendencies–where it tends to focus, where attention is first given–and this affects how people place importance on the subjects in a photograph. Imagine a grid drawn over a photograph, splitting it into nine areas. The four spots where these lines intersect are where the eye tends to give the most important, especially the starting on the left side (the opposite is true in countries where they read right to left.) The next most important places are along the lines themselves.
Understanding this, you can frame your shots to place your subjects and the most visually interesting elements along these points and lines. Many modern cameras can display this grid in their LCD screen or view finder so you can use it as a guide. With training you can simply visualize the lines as you take the photographs.