The Right of Vengeance was a traditional matter of honor that was claimed when an individual sought revenge against another. In the Klingon Empire, any Klingon, even one who had received discommendation, could claim it.
After Duras murdered K'Ehleyr in 2367, Lieutenant Worf claimed Right of Vengeance against Duras. Initially, Duras rejected the challenge, but subsequently accepted it when Worf declared that K'Ehleyr was his mate. Worf then killed Duras in hand-to-hand combat. (TNG: "Reunion")
In early 2368, when it was discovered that the House of Duras had conspired to destroy the honor of the Mogh, Chancellor Gowron offered Worf the life of Toral in exchange for his family's misdeeds against the House of Mogh. Worf declined because Toral was a boy who had done Worf no harm, a decision he regretted a few years later. (TNG: "Redemption II")