The basic principle behind the rule of thirds is to imagine breaking an image down into thirds (both horizontally and vertically) so that you have 9 parts. As follows. As you're taking an image you would have done this in your mind through your viewfinder or in the LCD display that you use to frame your shot.
Examples of the Rule of Thirds
Best picture
This is my best image because it focuses on the object. It has the correct white balance and iso, also it isn't over exposed.
Worst picture
This is the worst image as it does not focus on the image I was aiming for. However, it does focus on the grass. This image also shows camera shake as you can see an object on the right hand corner.