The rule of thirds (more of a guideline, really) is a basic rule applying to shot composition for photography -- both still and moving (e.g. cinema, TV). It is supposed to make your shots more interesting, and works like so: Imagine two horizontal lines dividing the image into equal thirds, and then two vertical lines dividing the image into equal thirds. This, as you should now have realized, gives you an image with nine squares. Good.

The rule says that the more interesting locations to place objects in your shot are at the intersections of the lines. For instance, if you're taking a closeup of one person, and nothing else, you should put their eyes on the first horizontal line, with each eye on one of the two vertical lines, putting their eyes a third of the way up and a third of the way from each side of the frame.

However, like any rule, it is meant to be broken. Since it is a "rule" applying to something artistic and creative, by all means, break it if you feel so inclined. It just gives you a reasonable starting point for your shot.