The masses and emission-line region sizes of active galactic nuclei (AGNs) can be measured by "reverberation-mapping" techniques, and we use these results to calibrate similar determinations made by photoionization models of the AGN line-emitting regions. Reverberation mapping uses the light travel-time delayed emission-line response to continuum variations to determine the size and kinematics of the emission-line region. We compile a sample of 17 Seyfert 1 galaxies and two quasars with reliable reverberation and spectroscopy data, twice the number available previously. The data provide strong evidence that the broad-line region (BLR) size (as measured by the lag of the emission-line luminosity after changes in the continuum) and the emission-line width measure directly the central mass: the virial assumption is tested with long-term UV and optical monitoring data on NGC 5548. Two methods are used to estimate the distance of the broad emission-line region from the ionizing source: the photoionization method (which is available for many AGNs but has large intrinsic uncertainties) and the reverberation method (which gives very reliable distances but is available for only a few objects). The distance estimate is combined with the velocity dispersion, derived from the broad Hβ line width (in the photoionization method) or from the variable part (rms) of the line profile, in the reverberation-rms method, to estimate the virial mass. Comparing the central masses calculated with the reverberation-rms method to those calculated using a photoionization model, we find a highly significant, nearly linear correlation. This provides a calibration of the photoionization method on the objects with presently available reverberation data, which should enable mass estimates for all AGNs with measured Hβ line width. We find that the correlation between the masses is significantly better than the correlation between the corresponding BLR sizes calculated by the two methods, which further supports the conclusion that both methods measure the mass of the central black hole . Comparing the BLR sizes given by the two methods also enables us to estimate the ionizing EUV luminosity Lion, which is not directly observable. Typically it is 10 times the monochromatic luminosity at 5100 Å (Lv). The Eddington ratio for the objects in our sample is in the range Lv/LEdd∼ 0.001-0.03 and Lion/LEdd≈ 0.01-0.3.