  The Dark Energy Spectroscopic Instrument (DESI) is carrying out a 5-year
survey that aims to measure the redshifts of tens of millions of galaxies and
quasars, including 8 million luminous red galaxies (LRGs) in the redshift range
of $0.4<z<{\sim}\,1.0$. Here we present the selection of the DESI LRG sample
and assess its spectroscopic performance using data from Survey Validation (SV)
and the first 2 months of the Main Survey. The DESI LRG sample, selected using
$g$, $r$, $z$, and $W1$ photometry from the DESI Legacy Imaging Surveys, is
highly robust against imaging systematics. The sample has a target density of
605 deg$^{-2}$ and a comoving number density of $5\times10^{-4}\
h^3\mathrm{Mpc}^{-3}$ in $0.4<z<0.8$; this is a significantly higher density
than previous LRG surveys (such as SDSS, BOSS and eBOSS) while also extending
to $z \sim 1$. After applying a bright star veto mask developed for the sample,
$98.9\%$ of the observed LRG targets yield confident redshifts (with a
catastrophic failure rate of $0.2\%$ in the confident redshifts), and only
$0.5\%$ of the LRG targets are stellar contamination. The LRG redshift
efficiency varies with source brightness and effective exposure time, and we
present a simple model that accurately characterizes this dependence. In the
appendices, we describe the extended LRG samples observed during SV.
