We present Hubble Space Telescope/WFPC2 observations of a well-defined sample of 40 X-ray sources with X-ray fluxes above the detection threshold of the full 1 Ms Chandra Deep Field-South (CDF-S). The sensitivity and spatial resolution of our HST observations are sufficient to detect the optical counterparts of 37 of the X-ray sources, yielding information on their morphologies and environments. In this paper we extend the results obtained in our previous study on the 300 ks CDF-S X-ray data (Schreier et al.). Specifically, we show that the optical counterparts to the X-ray sources are divided into two distinct populations: (1) an optically faint group with relatively blue colors, similar to the faint blue field galaxy population, and (2) an optically brighter group, including resolved galaxies with average colors significantly redder than the corresponding bright field galaxy population. The brighter objects comprise a wide range of types, including early- and late-type galaxies, starbursts, and active galactic nuclei (AGNs). By contrast, we show that the faint blue X-ray population is most consistent with being predominantly type 2 AGNs of low to moderate luminosity, located at higher redshifts (z ~ 1-2). This conclusion is supported by luminosity function models of the various classes of objects. Hence, the combination of deep X-ray data with the high spatial resolution of HST are for the first time allowing us to probe the faint end of the AGN luminosity function at cosmologically interesting redshifts.