Hard X-ray surveys have uncovered a large population of heavily obscured AGNs. They also reveal a population of quasars with moderate obscuration at both visible and X-ray wavelengths. We use Chandra-selected samples of quasars from the ELAIS Deep X-ray Survey (EDXS) and the Chandra Deep Field-North to investigate the obscuration toward the nuclei of moderately obscured AGNs. We find an inverse correlation between the optical-to-X-ray flux ratio and the X-ray hardness ratio that can be interpreted as being due to obscuration at visible and X-ray wavelengths. We present detailed optical and near-infrared data for a sample of optically faint (R > 23) quasars from the EDXS. These are used to constrain the amount of rest-frame UV/optical reddening toward these quasars. It is found that optically faint quasars are mostly faint because of obscuration, not because they are intrinsically weak. After correcting for reddening, the optical magnitudes of most of these quasars are similar to the brighter quasars at these X-ray fluxes. Combining with gas column densities inferred from the X-ray observations, we consider the gas-to-dust ratios of the obscuring matter. We find that the quasars generally have higher gas-to-dust absorption than that seen in the Milky Way, similar to what has been found for nearby Seyfert galaxies. We consider the possible existence of a large population of X-ray sources that have optical properties of type 1 (unobscured) quasars but X-ray properties of type 2 (obscured) quasars. However, we find that such sources only contribute about 6% of the 0.5-8 keV X-ray background. Finally, we show that the observed distribution of optical-to-X-ray flux ratios of quasars at z > 1 is skewed to low values compared to the intrinsic distribution as a result of the fact that the observed-frame R-band light is emitted in the UV and is more easily obscured than hard X-rays.