The Dark Energy Spectroscopic Instrument (DESI) is carrying out a five-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 0.4 < z ≲ 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 two 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 × 10-4 h 3 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 ~ 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.