It's hard perhaps to give a general definition but I'll try. generally one speaks of the fractal dimension of a surface or of an object. One definition, involves what is called box counting. You enclose the object with a regular grid of boxes (a lattice say a cubic lattice) You then count how many boxes contain parts of the object (some will be empty). You do this for different box sizes. (the box size is L say) you the study the scaling behavior of the number N(L) corresponding to the number of boxes of size L which contain parts of the object. The formula should be something like N(L) ~ L^(-d_f) where d_f is the fractal dimension, so that if you have a normal object then, as L gets small d_f will simply be the dimesions of the space (e.g. a 3d object in 3d space will have N(L) ~ L^-3 for sufficiently small L.
Jacques Amar