Imagine the viewfinder of your camera. Now imagine two horizontal lines placed on it, so that they divide the shape into three equal sections. Now imagine another two lines, this time vertical ones. So your viewfinder (or image frame) is now divided into three sections, vertically and horizontally.
Use these lines to position your subject, or part of the subject for the most impact. The "rule of thirds" dictates that the eye will be drawn more to the areas of the frame where the lines intersect. This is never at the dead centre of the image. So, putting auntie Mabel right in the middle of your shot, with lots of space all around her, is a no-no. And, yet the dead centre of the frame is probably the most common place to position a subject.
The other no-no is to place an object too close to the edge of the frame. This can make the overall scene look unbalanced. Also, when placing people in a frame, if they are looking to one side, it often works best to give them "room" to look into in the frame. So try to avoid placing someone at the edge of the frame looking out at it - unless you want a "I'm about to bang my head on this brick wall" effect.
It's the same issue with horizons in landscape shots. Where do a lot of photographers put the horizon? In the middle of the frame, giving a perfectly even and balanced amount of sky and land area. Boring! When following the rule of thirds, the horizontal lines (or horizon lines) are one third up from the bottom or one third down from the top. Try that for landscape and you'll either get lots of rolling hills, or vineyard (or whatever the main subject) and a thin strip of sky, or huge expanses of billowing clouds, or fiery red sunsets, and a strip land. Either way it'll be a lot more powerful (and the choice often depends on the subject itself and what you are trying to convey).
In my beach scene (above) the yacht is placed close to the intersection of the top horizontal line and the right hand vertical line. The horizon is close to the top horizontal line. The front line of rocks starts close to the bottom horizontal line.
Let me know what you think - and send me some of your own examples and I'll post them!
My email address is on the comments page (click "comments" link, below this post).