This article is from the sci.fractals FAQ, by Michael C. Taylor and Jean-Pierre Louvet with numerous contributions by others.
A fractal is a rough or fragmented geometric shape that can be
subdivided in parts, each of which is (at least approximately) a
reduced-size copy of the whole. Fractals are "generally" self-similar
and independent of scale.
There are many mathematical structures that are fractals; e.g.
Sierpinski triangle, Koch snowflake, Peano curve, Mandelbrot set, and
Lorenz attractor. Fractals also describe many real-world objects, such
as clouds, mountains, turbulence, coastlines, roots, branches of
trees, blood vesels, and lungs of animals, that do not correspond to
simple geometric shapes.
Benoit B. Mandelbrot gives a mathematical definition of a fractal as a
set of which the Hausdorff Besicovich dimension strictly exceeds the
topological dimension. However, he is not satisfied with this
definition as it excludes sets one would consider fractals.
According to Mandelbrot, who invented the word: "I coined "fractal"
from the Latin adjective "fractus". The corresponding Latin verb
"frangere" means "to break:" to create irregular fragments. It is
therefore sensible - and how appropriate for our needs! - that, in
addition to "fragmented" (as in "fraction" or "refraction"), "fractus"
should also mean "irregular," both meanings being preserved in
"fragment"." (The Fractal Geometry of Nature, page 4.)