There’s nothing new about the rule of thirds — it’s almost a photographic cliche. Still, as a, well, rule of thumb there’s a good deal of sense in it. Let’s have a look.
One of the worst instincts of amateur photographers is to aim the camera directly at the main subject, as if it were game to be bagged. You can see this in society pages, like one in the back of a magazine I’m responsible for (I try to keep the section’s space to a minimum). The photographer’s strategy in these situations is just about always to line the subjects up in a grinning row facing the camera. You can see what I mean in the above image (I’ve replaced the people’s faces with smilies so as not to embarrass anyone, and to highlight the composition).
The rule of thirds says that you’re better off arranging your composition with a main element a third of the way from one of the edges. In effect you imagine your image as composed of nine equal rectangles. Consider this image from the Sentiero degli Dei in the Lattari Mountains above Amalfi.
You can see that the cliff at the left is a third of the way in from the left edge of the photo. (You can also think of each of the nine squares as a section to be balanced in its own right.)
Or look at this photo from the Sentiero della Republica in the same region.