The rule of thirds is a photography technique to make you pictures be more engaging. You picture is divided into a 3×3 grid and placing your subject at a certain spot. Thus is to make your picture be more interesting and not boring placing everything at the center.
To use the rule of thirds you need to understand where to place your subject and horizon lines. It is pretty self explanitory with horizon becuase you just put it at one of the two lines but for the vertical lines it can be more complicated. Usually you want to put you subject in the top left of you image. When it comes portaits of people or animals you would want to put there eye on of the the intersection points.
The chart here shows which spots people look first.
All of these pictures are using the rule of thirds if it is there eye or the placement or the horizon but all of these images in some way are using the rule of thirds.
This is my featured image. I set this as my featured image because it is a great example of the rule of thirds with his eye being at the left intersection point.