Massive foreground galaxy clusters magnify and distort the light of objects behind them, permitting a view into both the extremely distant and intrinsically faint galaxy populations. We present here the z ~ 6 - 8 candidate high-redshift galaxies from the Reionization Lensing Cluster Survey (RELICS), a Hubble and Spitzer Space Telescope survey of 41 massive galaxy clusters spanning an area of ~200 arcmin^2. These clusters were selected to be excellent lenses and we find similar high-redshift sample sizes and magnitude distributions as CLASH. We discover 321 candidate galaxies with photometric redshifts between z ~ 6 to z ~ 8, including extremely bright objects with H-band magnitudes of m_AB ~ 23 mag. As a sample, the observed (lensed) magnitudes of these galaxies are among the brightest known at z> 6, comparable to much wider, blank-field surveys. RELICS demonstrates the efficiency of using strong gravitational lenses to produce high-redshift samples in the epoch of reionization. These brightly observed galaxies are excellent targets for follow-up study with current and future observatories, including the James Webb Space Telescope.