This article is from the sci.fractals FAQ, by Michael C. Taylor and Jean-Pierre Louvet with numerous contributions by others.
It is not easy to give a succinct definition of multifractals.
Following Feder (1988) one may distinguish a measure (of probability,
or some physical quantity) from its geometric support - which might or
might not have fractal geometry. Then if the measure has different
fractal dimension on different parts of the support, the measure is a
Hastings and Sugihara (1993) distinguish multifractals from
multiscaling fractals - which have different fractal dimensions at
different scales (e.g. show a break in slope in a dividers plot, or
some other power law). I believe different authors use different names
for this phenomenon, which is often confused with true multifractal