Look at almost anything: the distribution of patchy grass on a field, waves in the sea, the movements of an ant, the movement of branches of a tree, patterns in marble, winds. All these phenomena exhibit the same pattern of large and small variations. The Perlin Noise function recreates this by simply adding up noisy functions at a range of different scales.The following graphic was generated using PERLIN NOISE. All of the land, all the clouds and the water with its waves - all were produced by using perlin noise techniques.
In 1997, Ken Perlin received an academy award for developing the Perlin Noise generator. Perlin Noise has a multitude of uses ranging from the creation of natural textures (see a Texture demo) to artificial terrain and even worlds (see a Planet demo).
Lots of examples to check out: NYU home page: http://mrl.nyu.edu/~perlin/
Fractal noise is not unique to After Effects, the concept was developed for Hollywood in the early 80s by a remarkable guy called Ken Perlin. In many fields of graphics fractal noise is called "Perlin Noise", and a quick Google search will reveal lots of information on the principle, and how it has been implemented in everything from blockbuster films to computer games. A famous early use of fractal noise was in the terraforming scenes in "Star Trek II", and Ken Perlin eventually won an Oscar recognizing his efforts in 1997.