The "Rule of Thirds" is one of the first lessons photographers learn when studying composition and I consider whenever I lift my camera to my eye to photograph a puppy, child, family or a vacation photo. It is the basic principle of breaking an image down into thirds like a tic-tac-toe board so that you have 9 parts. When framing your image in the viewfinder, consider placing points of interest on an intersection of both the horizontal and vertical lines, also called "power points".

The reasoning behind placing important parts of your image at the intersections, or along the lines, is to create a more balanced image that allows the viewer to interact with it more naturally.  Studies have shown that viewers' eyes tend to go to one of the power points rather than the center of the photograph so using the Rule of Thirds actually reduces tension that a centered subject make create and makes it more pleasing to view.  

When using the Rule of Thirds (and don't forget, rules are also meant to be broken, so this is just one consideration in creating a beautiful image), you should always start with this question; "What is the subject of my photograph?"  It may be a puppy, person, mountain or tree, it doesn't matter.  Where do you want the viewer to look? Once you've answered that, you can frame your subject accordingly and play around with which power point works best for your image.

Let's start by looking at the image below, I've centered the puppy on the grid; meh.  It doesn't do much for me.  Cute puppy yes, but compositionally it's not very dynamic.