Nb superconducting radio-frequency (SRF) cavities are observed to break down and lose their high-Q superconducting properties at accelerating gradients below the limits imposed by theory. The microscopic origins of SRF cavity breakdown are still a matter of some debate. To investigate these microscopic issues, temperature- and power-dependent local third-harmonic response is measured on bulk Nb and Nb thin-film samples using a novel near-field magnetic microwave microscope between 2.9 and 10 K and 2 and 6 GHz. Both periodic and nonperiodic response as a function of applied rf field amplitude are observed. We attribute these features to extrinsic and intrinsic nonlinear responses of the sample. The rf-current-biased resistively shunted junction (RSJ) model can account for the periodic response and fits very well to the data using reasonable parameters. The nonperiodic response is consistent with vortex semiloops penetrating into the bulk of the sample once sufficiently high rf magnetic field is applied and the data can be fit to a time-dependent Ginzburg-Landau (TDGL) model of this process. The fact that these responses are measured on a wide variety of Nb samples suggests that we are capturing the generic nonlinear response of air-exposed Nb surfaces.