We have performed a successful targeted search for a population of red, radio-quiet, and probably absorbed quasars. Radio-quiet, optically red ROSAT PSPC X-ray sources brighter than 1 × 10-13 ergs cm-2 s-1 were searched for red (O-E > 2.0, O ≤ 20) counterparts in the Automated Plate-measuring Machine catalog of Palomar Sky Survey objects. Of 45 objects for which we obtained adequate follow-up optical spectroscopy, we have found seven red quasars, five with αopt < -2. Their redshifts range from 0.06 to 0.31, and their luminosities are moderate, lying on the quasar/Seyfert boundary. These red quasars strengthen the case for a radio-quiet population that is the counterpart of the radio-loud red quasars found by Smith & Spinrad and Webster et al. Unidentified, fainter sources could increase the fraction of red quasars by up to a factor of 7. For the red quasars found here, the Hα/Hβ ratios, optical slope, and X-ray colors all indicate that they are absorbed by AV~2 rather than having intrinsically red spectra. This amount of obscuration seems to hide ~1%-7% of quasars at a given observed flux or ~3%-20% when their fluxes are corrected to their intrinsic values. This size of population is consistent with earlier limits, with predicted values from Comastri et al., and is comparable with the rate found among radio-loud quasars. A large population of more heavily absorbed (AV=5), fainter quasars equal in size to the blue population could exist without violating existing upper limits, which is in accord with the Comastri et al. predictions.