The 'Rule of Thirds' is the process of placing objects in an image on invisible 'power lines'. This technique is beneficial when looking to include more of a background/foreground, whilst still keeping the main focus in the image.
This technique is used mainly with a deep depth of field, as the background/foreground is in focus as well as the main focal point.
The image below shows how the 'Rule of Thirds' works.
My Experimentation
The photographs below show my experimentation with the 'Rule of Thirds'. I imagined the vertical and horizontal lines going across my view finder as I took these images.