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The golden ratio is considered to be aesthetically. Mathematically, it's an irrational constant, aproximately 1.6180339887. It is denoted as the Greek letter phi (looks like a lower case "p" with a tail.)

The golden rectangle is my favorite example of the the golden ratio. It is created by first drawing a rectangle where the length is 1/3 the heigth. Next, the top third of the heigth is separated (with an imaginary line) and the bottom 2/3 are in half with a line on the smaller part. If done correctly, each of the smaller rectangles are exactly similar to the original, thus creating an infinate amount of rectangles.

At least since the Renaissance, many artists and architects have proportioned their works to approximate the golden ratio.