Rule of Thirds
Let's start right off with an example of a photo that uses the Rule of Thirds:
The rule of thirds basically means imagining your frame in 9 equal squares (2 horizontal and 2 vertical lines). The idea is that you place the subject of your photo on one of the intersecting lines. Like so:
Why should you follow this rule? Because the human brain generally feels better with this composition as it can add more excitement/interest to the photo. Centering the subject tends to have the opposite effect, making the photo uninteresting to the eye.
Now that I've told you to use the Rule of Thirds, I'm going to backtrack and say that it's not really a rule but more a compositional guideline. Nonetheless, this particular guideline is an important one to follow, especially when starting out. In the end, the only rule in art is that there are no rules.
Exceptions to the Rule...
The photo above was an 8x10 crop. To my eyes, the 4x6 ratio below actually looked better and I opted to follow my own rules:
Sometimes, the "subject" is interpreted differently by each person. Are the faces the main subject, or are their whole bodies the main subject? If their bodies, this image doesn't necessarily follow the rule of thirds but it still works in this case:
Anyway, that's about all there is to it. If you have further questions, shoot me an email.