We present the results of our spectroscopic follow-up program of the X-ray sources detected in the 942 ks exposure of the Chandra Deep Field-South (CDFS). A total of 288 possible counterparts were observed at the VLT with the FORS1/FORS2 spectrographs for 251 of the 349 Chandra sources (including three additional faint X-ray sources). Spectra and R-band images are shown for all the observed sources and R - K colors are given for most of them. Spectroscopic redshifts were obtained for 168 X-ray sources, of which 137 have both reliable optical identification and redshift estimate (including 16 external identifications). The R < 24 observed sample comprises 161 X-ray objects (181 optical counterparts), and 126 of them have unambiguous spectroscopic identification. There are two spikes in the redshift distribution, predominantly populated by type 2 active galactic nuclei (AGNs) but also type 1 AGN and X-ray normal galaxies: the one at z = 0.734 is fairly narrow (in redshift space) and comprises two clusters/groups of galaxies centered on extended X-ray sources, the second one at z = 0.674 is broader and should trace a sheetlike structure. The type 1 and type 2 populations are clearly separated in X-ray/optical diagnostics involving parameters sensitive to absorption/reddening: X-ray hardness ratio (HR), optical/near-IR color, soft X-ray flux, and optical brightness. Nevertheless, these two populations cover similar ranges of hard X-ray luminosity and absolute K magnitude, thus trace similar levels of gravitational accretion. Consequently, we introduce a new classification based solely on X-ray properties, HR, and X-ray luminosity, consistent with the unified AGN model. This X-ray classification uncovers a large fraction of optically obscured, X-ray-luminous AGNs missed by the classical optical classification. We find a similar number of X-ray type 1 and type 2 QSOs [LX(0.5-10 keV) > 1044 ergs s-1] at z > 2 (13 sources with unambiguous spectroscopic identification); most X-ray type 1 QSOs are bright, R ≲ 24, whereas most X-ray type 2 QSOs have R ≳ 24, which may explain the difference with the CDFN results as few spectroscopic redshifts were obtained for R > 24 CDFN X-ray counterparts. There are X-ray type 1 QSOs down to z ~ 0.5, but a strong decrease at z < 2 in the fraction of luminous X-ray type 2 QSOs may indicate a cosmic evolution of the X-ray luminosity function of the type 2 population. An X-ray spectral analysis is required to confirm this possible evolution. The red color of most X-ray type 2 AGNs could be due to dust associated with the X-ray absorbing material and/or a substantial contribution of the host galaxy light. The latter can also be important for some redder X-ray type 1 AGNs. There is a large population of EROs (R - K > 5) as X-ray counterparts, and their fraction strongly increases with decreasing optical flux, up to 25% for the R ≥ 24 sample. They cover the whole range of X-ray hardness ratios, comprise objects of various classes (in particular a high fraction of z ≳ 1 X-ray absorbed AGNs, but also elliptical and starburst galaxies) and more than half of them should be fairly bright X-ray sources [LX(0.5-10 keV) > 1042 ergs s-1]. Photometric redshifts will be necessary to derive the properties and evolution of the X-ray selected EROs.