We present the Complete Local-Volume Groups Sample (CLoGS), a statistically complete optically selected sample of 53 groups within 80 Mpc. Our goal is to combine X-ray, radio and optical data to investigate the relationship between member galaxies, their active nuclei and the hot intra-group medium (IGM). We describe sample selection, define a 26-group high-richness subsample of groups containing at least four optically bright (log LB ≥ 10.2 LB⊙) galaxies, and report the results of XMM-Newton and Chandra observations of these systems. We find that 14 of the 26 groups are X-ray bright, possessing a group-scale IGM extending at least 65 kpc and with luminosity >1041 erg s-1, while a further three groups host smaller galaxy-scale gas haloes. The X-ray bright groups have masses in the range M500 ≃ 0.5-5 × 1013 M⊙, based on system temperatures of 0.4-1.4 keV, and X-ray luminosities in the range 2-200 × 1041 erg s-1. We find that ∼53-65 per cent of the X-ray bright groups have cool cores, a somewhat lower fraction than found by previous archival surveys. Approximately 30 per cent of the X-ray bright groups show evidence of recent dynamical interactions (mergers or sloshing), and ∼35 per cent of their dominant early-type galaxies host active galactic nuclei with radio jets. We find no groups with unusually high central entropies, as predicted by some simulations, and confirm that CLoGS is in principle capable of detecting such systems. We identify three previously unrecognized groups, and find that they are either faint (LX, R50042 erg s-1) with no concentrated cool core, or highly disturbed. This leads us to suggest that ∼20 per cent of X-ray bright groups in the local universe may still be unidentified.
