  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.
