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/Ly{alpha}, NV/CIV, SiIV/CIV and other flux ratios, and very broad and blueshifted [OIII] {lambda}5007. Here we present a new catalogue of CIV and NV emission-line data for 216188 BOSS quasars to characterize the ERQ line properties further. We show that they depend sharply on UV-to-mid-IR colour, 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{AA} at redshifts 2.0-3.4. A broader search finds 235 more red quasars with similar unusual characteristics. The core ERQs have median luminosity <logL(ergs/s)>~47.1, sky density 0.010deg^-2^, surprisingly flat/blue UV spectra given their red UV-to-mid-IR colours, 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.