This is termed a power-law relationship. If the slope and dimension of the line is integer, then we are in the linear world. A slope of two, for example, indicates a proess where Y is proportional to X squared. Cantor Sets and other related data types have fractional slopes and dimension. For example, Y might be proportional to X to the 3.45 power.
Fractal dimension implies scale invariance or self-similarity.
If you look at a process at one scale, you see the same thing if you change the scale. Fractals imply nonlinearity in the process which created the fractal process. Fractals turn out to be one way to measure chaotic systems. By themselves, fractals do not have any unique cause or source.