New optical spectra of radio and optically selected quasars covering a wide range in luminosity have revealed several effects associated with the redshift differences between their strong emission lines. The range of the difference between the Mg II λ2800 and C IV λ1549 lines exceeds 4000 km s-1 in a sample of high-luminosity, optically selected objects, which is significantly larger than previously detected values of this difference (∼600 km s-1). A systematic redshift of C IV λ1909 to C IV λ1549 has also been found in these spectra, exceeding 2000 km s-1 in one high-luminosity object, and is found to correlate in a linear fashion with the larger redshift difference between the Mg II and C IV lines. A linear relation between the magnitude of the Mg II-C IV redshift difference and the inverse of the C IV equivalent width is also found. Direct comparison of the Mg II-C IV redshift difference with the continuum absolute magnitude measured at the position of C IV suggests, but does not demonstrate, a correlation between the magnitude of this redshift difference and luminosity. A sample of broad-absorption line (BAL) QSOs is found to have unusually large (∼1700 km s-1) Mg II-C IV redshift differences, and to deviate from the correlation between this difference and the inverse of the C IV equivalent width for the normal QSOs. These results are found to be consistent with the model of Kallman and Krolik (1986) in which the opacity source responsible for the redshift differences is electron scattering and the nature of the implied radial motion is infall. The results also suggest a stratified ionization structure of the quasar broad-line region, consistent with the results for the broad-line regions of certain Seyfert galaxies based on emission-line variability.