The very first time I went out with a camera and took pictures using a film roll with 36 pictures, only 14 pictures came out as I had hoped. My creative vision was too much for an old Yashica to handle.
Even then, as much as I like symmetry, I noticed that I prefer the subject being on a side rather than in the middle. As it turns out, this is actually a well-known technique to improve the aesthetics of photography.
So rule of thirds. As Wikipedia tells us, it is used in more than just photography. It works for cinematography as well. For photography, it is actually one of the easiest things to do and something I tell other people most often.
You know those grids some camera have which look as follows:
So the rule of thirds says that you should keep the important compositional element or the subject in the picture along these lines, and preferably on those green dots (the intersection of lines).
That is it.
Here are some examples:
You can also use rule of third to make pictures of landscape be awesome.
The following picture from Wikipedia gives a nice example of the difference this makes:
The important thing to notice is that you do not have to follow the lines exactly, but rather take them as a guideline.
The pictures below include pictures which do not have the subject EXACTLY on the line, but still, look cool.