The first thing a photographer has to realize is, not everything should be centered in the frame. Artists from the early days of canvas and oil painting have realized this and have since followed a simple rule, a rule called the "Rule of Thirds". This rule states that you should place object near invisible lines, found at the 1/3 mark of the respective image or at their intersecting points.

I traditionally try placing my images along the lines, or place the key element in the photo at the intersecting lines (such as a hand, or a stare).

You can see 3 examples of this rule in the supporting photos.

Another thing that beginners should start looking for are geometric shapes and lines. For some strange reason, humans have a subconscious liking to geometrical shapes.

Typically these shapes help direct the viewer's eye to the central item, or help tell the story for the picture. The green lines help you see this more clearly. The last photo below has both red "third" lines and the green "geometric" lines. Combining different photographic principles will quickly improve most photographical endeavors.

By all means, rules can be broken: they shouldn’t be "worshiped". If an object is better centered, then by all means, do so.

The best tip for any budding photographer is to simply keep your eye open to other people's work. Analyze their images and see, "what makes it so cool/good?"