The basic principle behind the rule of thirds is to imagine breaking an image down into thirds (both horizontally and vertically) so that you have 9 parts. As you’re taking an image you would have done this in your mind through your viewfinder or in the LCD display that you use to frame your shot.
The golden ratio, represented by the Greek letter 'phi', is said to be is a mathematical connection between two aspects of an object.
It can be artificially used – for example, some 20th century artists used it for the rectangular shape of their portraits from the long side to the short side.