The Hausdorff dimension is used to define the dimension of fractals, for example, the dimension of the Sierpinski triangle is log(3)/log(2).

To find the d-dimensional Hausdorff measure of a set: cover the set with very small balls, sum the diameter to the power of d of each ball, and take the lim inf as the balls get smaller. For integer dimensions, the Hausdorff measure is equivalent to the Lebesgue measure. The Hausdorff dimension of a set is the point where the d-dimensional Hausdorff measure changes from infinity to zero, i.e. the dimension of a set is d^* if for d < d^* its measure is infinity and for d > d^* its measure is zero.

From the abstract for a paper by Dierk Schleicher:

… we construct a set E ⊂ â„‚ of positive planar measure and with dimension 2 such that each point in E can be joined to ∞ by one or several curves in â„‚ such that all curves are disjoint from each other and from E, and so that their union has Hausdorff dimension 1. We can even arrange things so that every point in â„‚ which is not on one of these curves is in E. These examples have been discovered very recently; they arise quite naturally in the context of complex dynamics, more precisely in the iteration theory of simple maps such as z → sin(z).