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The foundation of chaos theory was found by Edward Norton Lorenz in 1961 when rather then starting a weather simulation at the beginning he started it midway through using the numbers on the print out (which were rounded to three figures after the decimal point i.e 0.506127 became 0.506.) That small difference was enough to produce totally different results by the time the new program had reached the point where the old one ended.
Many chaotic systems exhibit the phenomenon of strange attractors which are almost identical states (velocity and position of a particle for example) which nonetheless never actually repeat.
Fractals (self-similarity of systems at varying scales) are much used in computer graphics, where a simple formula can, by repeated iteration, produce complex forms. They are derived from application of chaos theory. The Mandelbrot curve is a fractal which most people will be aware of.
The analysis of noise is also influenced by fractals.
The fern image below is a fractal, examination will show that successively smaller parts are morphed copies of the whole.
The word "fractal" comes from the fact that such curves have dimensions which are fractions of whole numbers.
In practical daily life, the main use of chaos theory in the late 1980s and early 1990s was for people who did not understand 2+2 to use chaos theory jargon to pick up other people who did not understand 2+2, while nerdy mathematicians futilely tried to tell both they were doing it wrong.
 See also
- Wikipedia on fractals
- James Gleick's website (author of Chaos: Making a New Science)
- The ultimate Mandelbrot zoom