Unless you are brand spanking new to photography, you will have heard about using the rule of thirds to compose your images. It's a useful guideline that helps you move away from having your subject slap bang in the middle of frame, by giving you a grid that shows you where to position your subjects. (If you haven't used that yet, I suggest you start by reading all about it here, then come back to this!)
If you are still reading this, then it means you are ready to move on from the rule of thirds! Yay!
The good news is there are more "guides" that we can use to compose our images, and these are called the golden compositions.
You might also see them called the golden mean, Phi, divine proportions and the fibonacci compositions, but they are all essentially the same thing - they are all compositional tools that use the "golden ratio" of 1 : 1.618 . This ratio is found everywhere throughout nature, and therefore (apparently) naturally pleasing to us humans. Essentially, we are taking this naturally occurring form, and using this as a guide to how to compose our images.
Here's an introduction to the golden compositions and what they look like:
The Golden Rectangle
The first (and easiest to use) is the golden rectangle. This grid looks very similar to the rule of thirds grid, but with one important difference - instead of dividing the frame into equal thirds, the lines of the golden mean are much closer to the centre.
The same principle as the rule of third applies here - you want to line up your frame so that you are putting important elements along the lines, or at the intersections. This is particularly useful if you have an image that is does not quite fit the rule of thirds, but you want to keep to that same structure. It's also the easiest golden composition to start out with, since it's the same idea as your basic rule of thirds.
The Golden Spiral
Yes, that was easy wasn't it? We are going to take a up a notch now, and get a bit more complicated :)
You can also use the golden mean not as a grid as shown above, but as a spiral - like this one below. This is the basically the golden rectangle after it has been divided by the golden ratio into a series of rectangles and squares, each one getting smaller than the last. You'll either see this shown as a spiral or as a a series of rectangles getting smaller and smaller (you can see these both on the grid below)
It doesn't matter which you use as they are both essentially the same thing.
You can use this as framework for where to place elements within the frame - you want the most important element on the smallest circle / smallest box - or at least with the smaller rectangles.
The following examples don't fit exactly, but remember, it's just a guide :)
The Golden Triangle
This is the one I find most difficult to use! The golden triangle again uses the golden mean - but this time divides the frame into triangles whose shape has the 1 : 1.618 ratio.
This one is particularly useful if you have any diagonal lines in the frame as you can line them up with the lines in the triangles, or try to position elements so they run along these lines, like in these:
Once again, it's not exact, but rather they follow the general guidelines. This is an important point - these are guides, and you never want to introduce a weird felling in a photograph just to make it "fit" a composition rule.
Playing with the Golden Compositions in Post Processing
The easiest way to learn about these golden compositions is to go and have a look at them yourself. You can find overlays of all these in Lightroom - simply select the crop tool, then press O repeatedly on your keyboard, and it will shuffle through all the compositional overlays! (I didn't find this out until I had been using Lightroom for over a year - maybe I'm just slow so I hope someone else didn't know that!) To change the orientation of the grid overlay, press SHIFT + O.
In later versions of Photoshop, you can also find these same overlays. With the crop tool selected, just go to the options tool bar at the top and select the drop down menu, and then the golden composition you want to use. Or you can use exactly the same shortcuts as with Lightroom: Press O to cycle through them and SHIFT + O to change the orientation. (These crop overlays are only available in earlier Photoshop versions, I think this kicked in at the last boxed version, CS6, so if you have earlier versions, don't waste time trying to find them!)
Go through some of your old photographs - particularly any that don't quite seem to "fit" using the rule of thirds - and try to see if you can apply any of the golden compositions to them, or whether you could have shot the image slightly differently in order to fit them.