Composition is probably the most important factor in improving one's photographic images. It can be the difference between a "snapshot' and a work of 'art'. Having our subject placed in the center of an image can sometimes look plain or boring, but placing the subject off center encourages our eye to move around the frame and tends to make the image more interesting and dynamic.
The 'rule of thirds' is a compositional tool that many types of artists use to achieve a more balanced image. To do this we divide our frame into thirds: two vertical lines and two horizontal lines. When we look through the viewfinder we can imagine these lines being there or in some cameras you can actually choose a setting where the lines are visible in your viewfinder and/or the lcd screen. Then a person can use these guides to arrange the main elements of the picture.
For example, if your subject is a tree or structure of some kind, place it on one of the two vertical lines to the left or right of center. If you have a horizon in the picture, place it on one of the two horizontal lines. Which one? If there is an interesting sky, I usually try to put the horizon on the bottom line to include more sky. If the sky is boring or the foreground is interesting, I tend to put the horizon toward the top line.
While this technique is referred to as a 'rule', I see it as a 'strong suggestion'. There are always exceptions to the rule. Sometimes, because of the subject matter or because of the photographers location, it is not always possible to use the 'rule of thirds'. I have seen pictures created by professional photographers where they have purposely tilted the horizon diagonally and it was a very dynamic photograph. Especially in fine art photography there are no rules because art is very subjective, so there is nothing wrong in trying different variations.
But the reality is, that most of the time, the 'rule of thirds' is an excellent technique to improve the quality of our photography. It helps a viewer to linger around when viewing an image and examine it. It also can provide a context for the subject giving the viewer more information. Moving different elements of a scene to the intersections of these imaginary horizontal and vertical lines can help one to create much more interesting photographs.
Here is an image that shows the 'rule of thirds' technique. I positioned the old house towards the left side of the frame and because I felt the sky had some interest, I put the horizon toward the bottom third.