Imaginary lines are drawn dividing the image into thirds both horizontally and vertically. You place important elements of your composition where these lines intersect. The intersections are known as Sweet Spots, or Power Points.
Proponents of the technique claim that aligning a subject with these points creates more tension, energy and interest in the composition than simply centering the subject would.
The main reason for observing the rule of thirds is to discourage placement of the subject at the center, or prevent a horizon from appearing to divide the picture in half.
As with most rules, it is made to be broken. This rule doesn't work on everything, but it generally adds more interest to a painting or photograph then if you placed the subject dead centre.
For example, in this picture I took in California in 2009, I placed the seagull in the bottom right sweet spot.
I purposely picked this photo to show you as it has a lot going on in the background. I positioned myself so that the boat in the background is NOT on a sweet spot. Then you would have competition in the photo for the viewer's attention. As the gull is closest to the viewer, it should be the only subject hitting a sweet spot.
That being said.....sometimes you do want to draw attention to something in the foreground and the background at the same time....this one was taken near Bonavista. You can use more than one sweet spot at a time, but don't use them all at once....you lose the effect!
And sometimes nature doesn't quite cooperate, so you can still create an interesting composition by "being close" to the sweet spots....these cone flowers were taken at Manitou Cliff Dwellings in Colorado in 2008....
And of course...always break the rules!!! When it makes sense of course (And when it's not illegal!) This is Brandywine Falls in British Columbia.