What is a center-weighted photo? Say you're taking a picture of...your dog. How do you take the shot - you point the camera, focus on the dog, and shoot, placing the dog smack in the middle of the frame.
Ok, so what does it mean to have a center-weighted photo? It means your picture is boring. It means your picture is ill-composed. It means people look at the picture and look away, without being drawn into the scene, without mustering up the smallest aww of interest for what you've captured.
Yea, it's that bad.
So why does everyone do it? Mainly, it's because people like symmetry -- It's comforting and soothing to the eye. If it's natural to arrange items on a shelf in a nice, neat, symmetrical order, why shouldn't it be natural to take a picture of them in the same way? Well, it is natural, but what's natural isn't always what's interesting. Photography has deep connections to art; until the 1850's most successful paintings were entirely center-weighted. Take a look at, say, the Mona Lisa. See? She's right there, in the middle!
Sadly, those aren't the only reasons. Early cameras weren't so great at dealing with parallax, so what a photographer saw through the viewfinder wasn't what was recorded on the film. To compensate photographers left plenty of space around their subject, just in case. In fact, Kodak film historically came with an insert suggesting that you take pictures like this, to keep from accidently cropping of the head of, say, your mother. Of course, camera manufacturers licked the parallax problem decades ago, but the process of leaving space around the subject has endured.
Why, then, is it so freakin' awesome to move the subject out of the center? Graham King, an author of several books on photography, said "This inadvertent cropping of the subject can have a curious effect; it frustrates our expectations and forces us to imagine what might be taking place outside the frame. It’s the photographic parallel of a fleeting glimpse when we ‘catch something out of the corner of our eye.’ If we’re unable to take the second look we must use the image we retained to help complete the picture by conjecture."
Of course this rule, like any other, is made to be broken, but you can't break it properly until you're familiar with it, and understand the reasons for following the rule in most situations.