In my opinion, the rule of thirds is the single most important part of composing a solid photograph. The idea is that if you divide your frame into three sections horizontally and three sections vertically, you'll be left with a grid. Points of interest should be placed at the intersecting points of your imaginary lines.
Here's an example using my latest photo:
Now this of course doesn't have to be exact but try to follow it as closely as possible. You can see that I clearly intended to have the two tallest buildings anchor the image and they're anchoring the image because of where they're positioned in the frame. Everything else just falls into place around those two points. I'm also using lines to lead your eye through the image (notice how Park Avenue moves you all the way through to the sunset and Mt. St. Helens). That's a different point altogether though.
You can use this with portraits as well. Always aim to have your subject's eyes at one of the four intersecting corners. Never put them in the center! If anything, try to position main elements along the "thirds lines." This holds especially true for horizons. A dead-center horizon is a dead image. I guarantee you'll come away with much better photographs if you use this technique.
Some cameras even have grid displays built in to the viewfinder. If you have that option, turn it on and keep it on. I always have mine displayed as it helps me guide my compositions and I move quicker as a result.
