The Rule of Thirds
The rule of thirds is the first tool you come across when learning about composition. It's a technique which has been used by painters for centuries to make good use of the available area.
Look through your camera's viewfinder. Remember that what you see through the viewfinder of an slr camera is exactly the same as what would be printed in terms of size. This is called the frame, i.e. the image fills the frame.
Now split the frame by drawing two lines vertically and horizontally, so that 9 squares are formed. These lines must be equally spaced. Notice how there are four points where lines cross. These are points of interest.
If you place an element at one of these points, the eye will be drawn to it naturally. If you come from a background in taking holiday snapshots, you're probably used to putting the subject slap bang in the centre of the image. This makes the subject stand out, but it doesn't make for interesting photos as there's no room for anything else to complement it.
The lines themselves are important as well, not just the intersections. Use lines to draw the eye into the photo, and to divide various elements. For example if taking a photo of a person, put them along one of the lines rather than in the middle and see what happens. Give the subject some space.
The rule of thirds in practice
Let me show you some of my photos with this grid overlayed.
Now, this photo is by no means technically perfect or particularly amazing, but it does illustrate the idea of not putting the subject in the centre of the frame. Your eye is drawn to the pylon, but still wanders to the right to explore the valley. The horizon is contained entirely within the middle row, and the top of the pylon is near an intersection.
Is this more interesting that putting the pylon in the middle of the frame?
Let's look at another...
I like this photo, it feels nicely balanced. See how the boy watching is at one intersection and the horse is at the opposite one. The photo tells a story, though what that story is is up to your imagination.
The rule of thirds is a very useful technique to bear in mind when you take photos, but remember that it's more of a guideline than a rule - rules are meant to be broken, after all! That said, the point of this is to illustrate that the subject does not have to be in the centre of the frame, and often the most interesting photos follow the rule of thirds or something similar.
Give it a go, and once you feel you're comfortable, then try to break it!