We present a sample of 120 dust-reddened quasars identified by matching radio sources detected at 1.4 GHz in the FIRST survey with the near-infrared 2MASS catalog and color-selecting red sources. Optical and/or near-infrared spectroscopy provide broad wavelength sampling of their spectral energy distributions that we use to determine their reddening, characterized by E(B-V). We demonstrate that the reddening in these quasars is best-described by SMC-like dust. This sample spans a wide range in redshift and reddening (0.1 < z < 3, 0.1 < E(B-V) < 1.5), which we use to investigate the possible correlation of luminosity with reddening. At every redshift, dust-reddened quasars are intrinsically the most luminous quasars. We interpret this result in the context of merger-driven quasar/galaxy co-evolution where these reddened quasars are revealing an emergent phase during which the heavily obscured quasar is shedding its cocoon of dust prior to becoming a arcsecnormal arcsec blue quasar. When correcting for extinction, we find that, depending on how the parent population is defined, these red quasars make up < 15-20% of the luminous quasar population. We estimate, based on the fraction of objects in this phase, that its duration is 15-20% as long as the unobscured, blue quasar phase.