The ratio of nature
The golden ratio is the ratio between the elements of a form such that the sum of two elements are in the same ratio to the larger one, as are the larger and smaller elements to each other (see the rectangle below). This ratio approximates 1.618 (or 0.618; the two numbers are the reciprocals of each other) and is found throughout nature (for example in many seashells), art, architecture and also in the dimensions of the human body. It is also called the golden mean, golden number, golden section, golden proportion and divine proportion, and is closely linked to the Fibonacci Sequence (read more here) as the ratio of numbers in this sequence converges on the golden ratio.
Leonardo da Vinci and Piet Mondrian both incorporate the golden ratio directly in their paintings, and Stradivarius used it in the design and construction of his violins. Most famously, many architectural icons such as Stonehenge, Chartres Cathedral, the Great Pyramid of Giza and the Parthenon all exhibit the ratio in their geometric forms.
The aesthetics of ratios
The use of the golden ratio across these different fields probably owes more to our natural (unconscious) preference for the aesthetics of such forms than planned design. There is considerable psychological literature which supports the automatic (whether innate or learnt) liking that we all show for the golden ratio in many manifestations, although it may just reflect our more instinctual preference for familiarity, as so many forms in nature show the ratio in action. The golden ratio along with the Fibonacci sequence has also inspired discoveries in mathematics and other fields, most notably much of the work of Roger Penrose on crystal structures (which led to the development of Penrose Tiling).
The geometry of design
The golden ratio may just be design tradition, or a more fundamental natural preference, but it continues to influence designers today. In creating aesthetic designs, always consider use of the ratio as long as other important features are not compromised, as it will almost always create a more preferred aesthetic and other advantages too (read about the aesthetic-usability effect here and their practical applications on constructionskillstest.com).
Principles of Design, Revised and Updated by William Lidwell, Kritina Holden and Jill Butler (2010)