We analyze spectra for a large sample of 578 active galactic nuclei to examine the relationships between broad emission line properties and central supermassive black hole (SMBH) mass. We estimate SMBH masses by applying the virial theorem to the C IV λ1549 broad emission line. Although the FWHMs of C IV and Hβ appear nearly unrelated in individual objects, these FWHMs are well correlated when averaged over subsamples in our database. Therefore, the lines are equally valid indicators of the average SMBH mass in quasar samples. Our sample spans 5 orders of magnitude in SMBH mass, 6 orders of magnitude in luminosity, and a redshift range of 0 ≤ z ≤ 5. Most lines diminish in equivalent width with increasing black hole mass (the usual "Baldwin effect"), and there are no trends with redshift. Recent studies indicate that there is a relationship between SMBH mass and the overall bulge/spheroidal component mass of the surrounding galaxy. This relation, together with the well-known mass-metallicity relationship among galaxies, predicts a relationship between SMBH mass and quasar metallicity. We estimate the metallicity in the broad emission line region by comparing several line ratios involving nitrogen to theoretical predictions. We find that the data are consistent with a trend between SMBH mass and metallicity, with some line ratios indicating a very strong trend, but the uncertainties in several other important line ratios are too large to confirm or test this correlation.