Continued fractions appear naturally in the resolution of quotient singularities of surfaces (and presumably in higher dimensions as well). From a topological point of view, a neighborhood of the singularity is the cone on a lens space L(p,q), and a particular continued fraction for q/p gives an explicit piece of a smooth complex surface with the same boundary. This is explained very nicely in the notes, "Differentiable Manifolds and Quadratic Forms" by Hirzebruch, Neumann, and Koh, and presumably in many algebraic geometry texts.