This is the first release of optical spectroscopic data of low-redshift Type Ia supernovae (SNe Ia) by the Carnegie Supernova Project including 604 previously unpublished spectra of 93 SNe Ia. The observations cover a range of phases from 12 days before to over 150 days after the time of B-band maximum light. With the addition of 228 near-maximum spectra from the literature, we study the diversity among SNe Ia in a quantitative manner. For that purpose, spectroscopic parameters are employed such as expansion velocities from spectral line blueshifts and pseudo-equivalent widths (pW). The values of those parameters at maximum light are obtained for 78 objects, thus providing a characterization of SNe Ia that may help to improve our understanding of the properties of the exploding systems and the thermonuclear flame propagation. Two objects, namely, SNe 2005M and 2006is, stand out from the sample by showing peculiar Si II and S II velocities but otherwise standard velocities for the rest of the ions. We further study the correlations between spectroscopic and photometric parameters such as light-curve decline rate and color. In agreement with previous studies, we find that the pW of Si II absorption features are very good indicators of light-curve decline rate. Furthermore, we demonstrate that parameters such as pW2 (Si II 4130) and pW6 (Si II 5972) provide precise calibrations of the peak B-band luminosity with dispersions of {approx}0.15mag. In the search for a secondary parameter in the calibration of peak luminosity for SNe Ia, we find a {approx}2{sigma}-3{sigma} correlation between B-band Hubble residuals and the velocity at maximum light of S II and Si II lines.