In a survey of quasar candidates selected by matching the Faint Images of the Radio Sky at Twenty cm (FIRST) and Two Micron All-Sky Survey catalogs, we have found two extraordinarily red quasars. FIRST J013435.7-093102 is a 1 Jy source at z=2.216 and has B-K>~10, while FIRST J073820.1+275045 is a 2.5 mJy source at z=1.985 with B-K~8.4. FIRST J073820.1+275045 has strong absorption lines of Mg II and C IV in the rest frame of the quasar and is highly polarized in the rest-frame ultraviolet, strongly favoring the interpretation that its red spectral energy distribution is caused by dust reddening local to the quasar. FIRST J073820.1+275045 is thus one of the few low radio luminosity, highly dust-reddened quasars known. The available observational evidence for FIRST J013435.7-093102 leads us to conclude that it too is reddened by dust. We show that FIRST J013435.7-093102 is gravitationally lensed, increasing the number of known lensed, extremely dust-reddened quasars to at least three, including MG 0414-0534 and PKS 1830-211. We discuss the implications of whether these objects are reddened by dust in the host or lensing galaxies. If reddened by their local environment, then we estimate that between 10% and 20% of the radio-loud quasar population is reddened by dust in the host galaxy. The discovery of FIRST J073820.1+275045 and objects now emerging from X-ray surveys suggests the existence of an analogous radio-quiet red quasar population. Such objects would be entirely missed by standard radio or optical quasar surveys. If dust in the lensing galaxies is primarily responsible for the extreme redness of the lensed quasars, then an untold number of gravitationally lensed quasars are being overlooked.