We report extensive laser-induced damage threshold measurements on dielectric materials at wavelengths of 1053 and 526 nm for pulse durations $\ensuremath{\tau}$ ranging from 140 fs to 1 ns. Qualitative differences in the morphology of damage and a departure from the diffusion-dominated ${\ensuremath{\tau}}^{\frac{1}{2}}$ scaling of the damage fluence indicate that damage occurs from ablation for $\ensuremath{\tau}l~10$ ps and from conventional melting, boiling, and fracture for $\ensuremath{\tau}g50$ ps. We find a decreasing threshold fluence associated with a gradual transition from the long-pulse, thermally dominated regime to an ablative regime dominated by collisional and multiphoton ionization, and plasma formation. A theoretical model based on electron production via multiphoton ionization, Joule heating, and collisional (avalanche) ionization is in quantitative agreement with the experimental results.