The Rule of Thirds is one of the primary rules of photography. It is based upon dividing up the image into an equal grid of thirds, so that there are nine squares that make up one photo. The idea behind it is that the eye is naturally drawn to the points of intersection, thus you should aim to focus significant points at these intersections.
In the photo below, the girl and tree are points of interest and are placed at the points of intersection in the photo, creating a sense of balance.
If you are taking a close up of a person, you should aim to centre the points of interest, such as eyes or in the case below, their hand, on a point of intersection. As a viewer's eye is naturally drawn here, it will bring out the best in your photography, achieving balance and creating a unique point of interest, whilst avoiding awkward photos.
The lines can also function as useful guides on how to frame your photo. For example if you're photographing a horizontal landscape such as the ocean, position the intersecting line at the point between the sky and the ocean to give your photo balance. In the photo below, the photo is divide into three parts; the ocean, the mountains and the sky. The ship is also situated on an intersection point which illustrates how the rule of thirds works to create balance, improve composition and capture beauty.
Once you've learnt the rule and it becomes natural to your photography you can experiment with breaking it in interesting ways.