The photography rule of thirds is a composition technique not just for photography, but it is perhaps the most widely used in photography.
The rule of thirds simply involves dividing your camera frame into equally spaced areas both vertically and horizontally.
These imaginary lines are used to place elements in you composition close to these lines or at their intersecting points.
I chose three sample photos to illustrate the rule of thirds. The first two examples have been edited by doing a crop of the original photo.
The photo on the left of the underside of the ocean pier is the UN-edited original. This interesting perspective had a lot of depth which leads our eye into the photo. The distant opening at the end of the pier was placed in the middle of the photo.
In the photo on the right I edited by cropping and came in a little closer and moved the opening to one of those imaginary intersection points. The diagonal line coming inward from the top right corner became more prominent.
The resulting photo is quite different, even though I only changed the composition a small amount. The exposed sand that was view-able in the bottom of the original, acted as a framing devise and is now missing as a composition element.
In this photo of the sunset and its reflection in the water, one of those common photo mistakes was made in the original. That rule says you should never place the horizon at the middle of your composition. It will divide your photo in half.
The photo has a nice balance with the three composition elements of the water, the rocks and the sky.
I did two photo edits. In the middle photo I used the rule of thirds and placed the horizon at the one third level of the composition. This emphasizes the sky and the rocks more than the original.
The photo on the right is my personal favorite crop. I placed the horizon at the two thirds level. This emphasizes the patterns and reflections in the water which I think is the most visually interesting part of the photo.
Both of the adjustments to the compositions above were done by cropping. For an easy way to employ the photography rule of thirds in-camera, read this: Rule of Thirds in Photography.
Photography composition rules are subjective and should not be followed blindly. The best part of digital photo editing is that you can try a few variations.
One additional tip is remember to rename your variations when you are saving them on your computer so that you won't overwrite and lose your original.
I did not do any editing on this composition of the arch reflection. I think it's a good example of not centering the oval of the archway vertically in the photograph. It's not a true use of the photography rule of thirds, but I think by now you get the idea.
The biggest composition mistake when photographing reflections is to place the horizon in the middle of the photo.
You can always do a rule of thirds crop when you do your photo editing afterward if you don't get your photography composition right in your camera frame.
Of course it saves you time if you get it right when you take your photograph.
Best wishes with the rule of thirds!