The Golden Ratio has many other names. You might hear it referred to as the Golden Section, Golden Proportion, Golden Mean, phi ratio, Sacred Cut, or Divine Proportion. They all mean the same thing.
In its simplest form, the Golden Ratio is 1:phi. This is not pi as in π or 3.14... and is not pronounced "pie." This is phi and is pronounced "fie."
Phi is represented by the lower-case Greek letter φ. Its numeric equivalent is 1.618...which means its decimal stretches to infinity and never repeats (much like pi). "The DaVinci Code" had it wrong when the protagonist assigned an "exact" value of 1.618 to phi.
Phi also performs amazing feats of derring-do in trigonometry and quadratic equations. It can even be used to write a recursive algorithm when programming software. But let's get back to aesthetics.
What the Golden Ratio Looks Like
The easiest way to picture the Golden Ratio is by looking at a rectangle with a width of 1, and a length of 1.168... If you were to draw a line in this plane so that one square and one rectangle resulted, the square's sides would have a ratio of 1:1. And the "leftover" rectangle? It would be exactly proportionate to the original rectangle: 1:1.618.
You could then draw another line in this smaller rectangle, again leaving a 1:1 square and a 1:1.618... rectangle. You can keep doing this until you're left with an indecipherable blob; the ratio continues on in a downward pattern regardless.