The Right of Vengeance was a Klingon right that was invoked over a matter of honor where one warrior seeks revenge against another. Any Klingon, even one who has been discommendated, may claim the right.
The claimant engages in mortal combat against his enemy until death, or - where the Klingon High Council has convicted said enemy - claims the right to pronounce and carry out sentence, usually executing the convict. However in that case, the claimant also has the right to show clemency to his foe and spare his life. (TNG episodes: "Reunion", "Redemption", DS9 episode: "The Sword of Kahless")