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 4 optically bright ( log L _ { B } \geq 10.2 ~ { } \mathrm { L _ { B \odot } } ) 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 > 10 ^ { 41 } \mbox { ~ { } erg } \mbox { ~ { } s } ^ { -1 } , while a further 3 groups host smaller galaxy-scale gas haloes . The X-ray bright groups have masses in the range M _ { 500 } \simeq 0.5-5 \times 10 ^ { 13 } \mathrm { ~ { } M _ { \odot } } , based on system temperatures of 0.4-1.4 keV , and X-ray luminosities in the range 2-200 \times 10 ^ { 41 } \mbox { ~ { } erg } \mbox { ~ { } s } ^ { -1 } . We find that \sim 53-65 % of the X-ray bright groups have cool cores , a somewhat lower fraction than found by previous archival surveys . Approximately 30 % of the X-ray bright groups show evidence of recent dynamical interactions ( mergers or sloshing ) , and \sim 35 % of their dominant early-type galaxies host AGN 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 ( L _ { X,R 500 } < 10 ^ { 42 } \mbox { ~ { } erg } \mbox { ~ { } s } ^ { -1 } ) with no concentrated cool core , or highly disturbed . This leads us to suggest that \sim 20 % of X-ray bright groups in the local universe may still be unidentified .