The Fibonacci sequence is a sequence of numbers made by starting with 1,1 and summing the previous two numbers in the sequence to produce the next number: 1,1,2,3,5,8,13...

This sequence appears often in Nature, see http://www.world-mysteries.com/sci_17.htm and http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html for examples, and has been posited as a mystery requiring the existence of a designer, or God (http://www.youtube.com/watch?v=-Ibc8sD5sgw) .

The reason so-called perfect rectangles contain squares with sides equal to the Fibonacci sequence is not that mysterious once you remember that each number in the sequence is the sum of the previous two, meaning you can always fit the two previous numbers in a length equal to the number after them.

The reason many spirals in Nature bear a vague resemblance to the Fibonacci spiral (and the similarity is rather vague --look at second 2:39 in the above video to see that the natural spiral falls shy of the Fibonacci one some places and extends beyond it in other) is that it's the most compact spiral you can build with squares or circles (and many of nature's building blocks have square-like or circular symmetry), as the least you can extend if you want to spiral around something is the return trip of what you've done in the opposite direction, which, if you keep turning directions every two pieces, is the sum of the lengths of the last two. Since many natural spirals such as shells need to fill in, they have a natural selective pressure to make the most compact spiral possible.

Other appearances of the Fibonacci sequence have more to do with self-similar generation patterns than with spirals. The Fibonacci pattern converges onto an exponential curve around the Golden ratio of 1.618034, meaning each number in the sequence approximates the previous number times the Golden ratio, and the sequence approximates the powers of the Golden Ratio. If you keep multiplying your numbers by the same number every unit time, you get an exponential like this. That's what biological populations and growing plants often do. Of all exponentials, the Fibonacci sequence is the simplest one with a gestational period of growth, namely the one for which the number of children born in any generation is proportional not to the current population size but to the population size one generation ago (i.e. those who have become adults now).