Rule of Thirds
Using the rule of thirds helps with photography in a major way, for example here is two of my photos using the rule of thirds.
The rule of thirds divides a photo into 9 equal parts.
Photos that have something hitting them in the vertical or horizontal line normally looks better, example the building the photo i took below, the line on the right travels down were the building meets the sky. It gives a nice amount of building to space. Also the middle 3 segments are were the light comes through the arc of the building which also leads the eye to it.
With the sharpenings in the photo i took below the rule of thirds helps in a different way, thhere is about the same amount of white space on top and bottom of the photo leaving the centre of the photo to hold the part of the image that we want the eye to focus too.
The Golden Ratio
The golden ratio works in a different way, helps explain where the eye leads to in the photo and what we want the viewer to focus on.
As you can see below the curve of the shape leads your eye into the lipstick that is closest to the camera lense.
Perhaps its easier to understand with the photo below i took of a church/chapel, there is a strange burst of light through the cloudy sky which i thought may symbolise a heaven above the place of worship for those who are religious. The golden ratio leads your eye exactly to this colourful burst in the sky and also has a large empty space for the church to fill with its dark silhouette.