Yes, all the experienced photographers know the "golden" rules of composition and use them almost innately. But until you start to use these simple rules without thinking, you need to practice, think what is better, and read about these rules. I am not going to describe the mathematical basics of these rules because they are well-described in other places, but here you'll find some examples, links to those descriptions, as well as JavaScript which could help you.
Rule of Thirds or Golden Ratio
So, the first "golden" rule is the "Rule of Thirds" or "Golden Ratio". It affects the ratio (1:1.618) of a picture size, as well as the placement of the main subjects in the photo. This ratio is close to the 35mm ratio, so you don't need to change the size of the photo in most cases. But you need to consider the composition: main subject should lie on one of the four lines or four intersections (subject's eye for example). Truthfully speaking, these rules are not the same. Rule of Thirds is a simplified version of the Golden Mean.
Golden Triangles
Another rule is the "Golden Triangles". It's more convenient for photos with diagonal lines. There are three triangles with corresponding shapes. Just roughly place three subjects with approximate equal sizes in these triangles and this rule would be kept.
Golden Spiral or Golden Rectangle
And one more rule is a "Golden Spiral" or "Golden Rectangle" (you'll see why it's a rectangle in the tools section). There should be something, leading the eye to the center of the composition. It could be a line or several subjects. This "something" could just be there without leading the eyes, but it would make its job.
