I subscribe to PictureCorrect, an online photography group. Periodically, they will send some really interesting information, and this article, The Rule of Thirds is one that anyone even remotely interested in photography, should read. It is only technical in the material presented, not the way it is written. The reader, I think, can easily grasp the concept. For instance, and from the article,

What is the rule of thirds? It is a sort of guideline to help you structure your composition in a balanced way that pleases the eye. Like all rules, it is made to be broken, but we will look at that later. Experience will give you the confidence to decide when you need to apply the rule of thirds, and when you can get by without it. However, before you can make that decision you need to understand the rule and how it works.

Imagine the rectangular shape of any photograph. Now draw a horizontal line one third from the top of the frame, and another one third from the bottom. Then draw a vertical line, one third from the left, and another one third from the right. Your rectangle should now be divided into nine equal sections.

According to the rule of thirds, the lines that divide the picture into thirds are the most effective places to position objects in your photo. So, for example, the horizon should be positioned on or near the line a third from the top or a third from the bottom of the picture. Vertical objects like trees should be placed on or near the lines a third from the left or right of the picture.

Also according to the rule, the most powerful points in the composition are the areas where the lines intersect. So, if your horizon is a third from the top of the frame, a house or tree on the horizon would be best placed a third from the left or right, at the intersecting point of the horizontal and vertical lines. If you have positioned a tree along one of the vertical lines, a bird sitting in a fork of the tree would be best positioned where it intersects with the horizontal line a third from the top.

There are some great photographs with the article that will visually give you an idea of what they are saying.