The rule of thirds is a composition guide far older than photography. Painters were using it. It is one of the basic composition rule. I know that all rules are meant to be broken, but to do that consciously, you have to learn them first.
You divide the screen in three equal parts vertically and horizontally. You want to place your major elements on those lines. The composition is even stronger if you put them at the crossing of those lines.
Let's see some examples. In the picture below, you see the difference between a centered composition vs one based on the rule of thirds. The one to the right has a much stronger composition. Notice that the horizon is on the lower line, dividing the middle and lower third. The family is on the node resulting from the crossing of the lower and right lines. That is where the most important part of the image is located.
In portrait, try to place the eyes on the upper line. If you place the nose on one of the vertical lines, it is even better as in the example below. You can also notice that the mouth is on the lower line.
It is even more important in landscape photography. Placing the horizon in the middle of your photo will be less pleasant to the eyes. In the following picture, the horizon is on the lower line, dividing the lower third from the rest of the image. The Frontenac Castle is on the left line and the gazebo with the Canadian flag is on the right line. That is a very welled composed image that is pleasant to look at. It draws the eyes to the important parts of the photo.
It is one of the basic rules of photography. It is also the same rule for cinematography. Learn it well. Practice it a lot. Then break it. :)