A fractal is a geometric shape that exhibits self-similarity across all scales.

The image being generated below is a well known example of a fractal, called The Koch Curve.
Each line segment is replaced by the original arrangement over a number of iterations (in this case
four iterations). As you can see in the last iteration, no matter how close you zoom into
the image, each portion represents the whole shape. If the iterations continued down to the infinitely
small, the self-similarity would still be maintained.

(Click on image to start iterations)

Theoretical fractals are infinitesimally subdivisble, such that each part contains no less detail than the whole. In reality, at some stage of subdivision, the detail will be lost. Hence it can be said that although true fractals cannot exist in reality, objects can possess fractal properties across a certain range of scale.