Chon-kar was a Narn blood oath, or vow of vengeance.
In Narn culture, when a Chon-kar was declared, the head of the family was honor-bound to see it through and, in the event that they are killed, the responsibility falls to the next-in-line.
Na'Toth's family had sworn a blood oath against the Dilgar war criminal Jha'dur. Na'Toth's grandfather and his family were on Hilak VII when the Dilgar invaded. Jha'dur used them in her illegal medical experiments, killing all but Na'Toth's grandfather who escaped back to Narn, but not before Jha'dur implanted a cybernetic device in his brain that would slowly kill his mind and spirit, leaving his surviving family helpless to do anything but watch.