We present a detailed X-ray spectral analysis of the sources in the 1Ms catalog of the Chandra Deep Field South (CDFS) taking advantage of optical spectroscopy and photometric redshifts for 321 extragalactic sources out of the total sample of 347 sources. As a default spectral model, we adopt a power law with slope Γ with an intrinsic redshifted absorption N_H, a fixed Galactic absorption and an unresolved Fe emission line. For 82 X-ray bright sources, we are able to perform the X-ray spectral analysis leaving both Γ and NH free. The weighted mean value for the slope of the power law is < Γ > ≃ 1.75 ± 0.02, and the distribution of best fit values shows an intrinsic dispersion of σ_int ≃ 0.30. We do not find hints of a correlation between the spectral index Γ and the intrinsic absorption column density N_H. We then investigate the absorption distribution for the whole sample, deriving the NH values in faint sources by fixing Γ = 1.8. We also allow for the presence of a scattered component at soft energies with the same slope of the main power law, and for a pure reflection spectrum typical of Compton-thick AGN. We detect the presence of a scattered soft component in 8 sources; we also identify 14 sources showing a reflection-dominated spectrum. The latter are referred to as Compton-thick AGN candidates. By correcting for both incompleteness and sampling-volume effects, we recover the intrinsic NH distribution representative of the whole AGN population, f(N_H) dN_H, from the observed one. f(N_H) shows a lognormal shape, peaking around log(N_H)≃ 23.1 and with σ ≃ 1.1. Interestingly, such a distribution shows continuity between the population of Compton-thin and that of Compton-thick AGN. We find that the fraction of absorbed sources (with N_H>1022 cm-2) in the sample is constant (at the level of about 75%) or moderately increasing with redshift. Finally, we compare the optical classification to the X-ray spectral properties, confirming that the correspondence of unabsorbed (absorbed) X-ray sources to optical type I (type II) AGN is accurate for at least 80% of the sources with spectral identification (1/3 of the total X-ray sample).