I learnt these compositional techniques in art class and they really relate to photography. There's a few out there but I probably use these the most and now I seem to apply them subconsciously. I tried placing the spiral on two images I'm editing in photoshop and they fit pretty well. So next time you're composing a photograph, have a think about using these techniques.
Fibonacci spiral or Golden Spiral:
The spiral was first discovered by Pythagoras in 5th Century B.C. It comes from the Fibonacci sequence: 1,1,2,3,5,8,13,21,34,55,89,144 etc. This is the result of adding the two previous numbers together to get the next number. To put it simply, the spiral is made from making a square from original rectangle which will leave a new rectangle. A new square is then cut from this leftover rectangle, leaving another rectangle to cut a square from and so on. The spiral can be found everywhere from nature to architecture to paintings. The idea in photography is that images are aesthetically pleasing to the eye when the focal point of the image lies under the centre of the spiral.
Rule of Thirds:
Pretty self explanatory the frame is divided into three equal parts, horizontally and vertically (the spiral is a little different to thirds). Our natural tendency is to frame whatever we’re taking a photograph of in the centre, where as if we place the point of focus over one or more of the point where the thirds meet then it is often creates a much more interesting composition.