I commend you for wanting to understand the rule of thirds. This is the number one most important rule of photography. Once you grasp the concept, your photos will linstantly improve, guaranteed.
Here's a rule of thirds chart. It divides the frame into thirds vertically and horizontally. Place the subject of your photo on this line, not in the middle of the frame. Images placed in the center of the frame seem "static," or unmoving, still. Subjects placed on the intersections of the lines, on the other hand, are "dynamic," or moving (which makes your photos more interesting and natural).
Notice his eyes form a line across the top line.
Look where the focus of the images is: near the top left intersection.
My friend's whole body is on the right hand side vertical line. Notice his head and shoes are near the grid's 4 intersection points.
First of all, notice the use of framing in this one. The foreground trees and grass make a border around part of the frame. Also notice that the foreground is parallel to the bottom line, and the top of the mountains in the background are near the top line.
Well that's all I've got for now. If you have any more questions just let me know.