There are several ways of defining a fractal, and a reader will need to reference their source to see which definition is being used.

Perhaps the simplest definition is to define a fractal to be a subset of with Hausdorff dimension greater than its Lebesgue covering dimension. It is worth noting that typically (but not always), fractals have non-integer Hausdorff dimension. See, for example, the Koch snowflake and the Mandelbrot set (named after Benoit Mandelbrot, who also coined the term “fractal” for these objects).

A looser definition of a fractal is a “self-similar object”. That is, a subset or which can be covered by copies of itself using a set of (usually two or more) transformation mappings. Another way to say this would be “an object with a discrete approximate scaling symmetry”.

See also the discussion near the end of the entry Hausdorff dimension.