Red quasars are candidate young objects in an early transition stage of massive galaxy evolution. Our team recently discovered a population of extremely red quasars (ERQs) in the Baryon Oscillation Spectroscopic Survey (BOSS) that has a suite of peculiar emission-line properties including large rest equivalent widths (REWs), unusual wingless line profiles, large NV/Lya, NV/CIV, SiIV/CIV and other flux ratios, and very broad and blueshifted [OIII] 5007. Here we present a new catalog of CIV and NV emission-line data for 216,188 BOSS quasars to characterize the ERQ line properties further. We show that they depend sharply on UV-to-mid-IR color, secondarily on REW(CIV), and not at all on luminosity or the Baldwin Effect. We identify a core sample of 97 ERQs with nearly uniform peculiar properties selected via i-W3 > 4.6 (AB) and REW(CIV) > 100 A at redshifts 2.0-3.4. A broader search finds 235 more red quasars with similar unusual characteristics. The core ERQs have median luminosity log L (ergs/s) ~ 47.1, sky density 0.010 deg^-2, surprisingly flat/blue UV spectra given their red UV-to-mid-IR colors, and common outflow signatures including BALs or BAL-like features and large CIV emission-line blueshifts. Their SEDs and line properties are inconsistent with normal quasars behind a dust reddening screen. We argue that the core ERQs are a unique obscured quasar population with extreme physical conditions related to powerful outflows across the line-forming regions. Patchy obscuration by small dusty clouds could produce the observed UV extinctions without substantial UV reddening.
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