In order to take a photo, you must first have to have a subject. The objects or people that you choose to photograph will need to be places strategically throughout the frame in order to create a compelling image. You can do this in by moving the camera right to left, or up and down, but also whether you want to be above or below your subject. If you shoot from below them, looking up, it generally displays your subject as powerful. On the contrast, if you shoot from above them, looking down, it will sometimes make the subject look small. It is equally important to consider how far away you must be from the subject. Those are some of the things to keep in mind when photographing. The "Rule of Thirds" is a guideline which applies to arranging a composition.

These lines can be displayed on your image through the viewfinder or on the screen of some cameras. The idea is to place the subject on one of the vertical lines and facing the direction of the other line. For example, notice how the subject in the following photo is on the right 3rd line and facing the open space in the image.

Or how this dog is looking from the left of the frame towards the right. Notice how the road also lines up with the higher horizontal 3rd line.

The goal is to be able to shoot the image lined up on the invisible lines so that you don't have to crop your image while you edit, although there are usually some minor tweaks that must be changed.
It is especially important to implement this technique when photographing moving subjects. The subject should be positioned on either the right or the left 3rd lines, and moving into the frame, or towards the other vertical line. Notice how the following subject has ample room in the frame to move around, the viewer has enough information to give a feel of what it is like to be present in that moment.

Or when looking at this image of a cowboy riding his horse. You can see what is in front of the horse and the landscape is clearly displayed. You can almost feel what its like to be in that empty, windy, and dry desert. Notice how the camera was low and turned up, to make the subject appear powerful and large.

This idea is also implemented in landscape and portrait photography. The 3rd lines are thought of as guidelines to composing. However, sometimes the composition looks best with the subject smack dab in the center.

Although, notice in the following photo that the main subject is centered, but the supporting subjects in the foreground are aligned to both vertical 3rd lines.

Arranging your composition is as simple as finding subjects, putting them on whichever 3rd line best represents them and the situation, deciding how far away you want to be, if you want to be above or below them, and your composition is done.
This is a nice way of showing how to shoot a picture properly with the right lighting, angle, and with the use of the Grid mode of a Camera.