vengeance: punishment inflicted in return for a wrong, by one who has the standing to inflict it
This noun names vengeance, the punishment a wrong calls down on the one who did it. The striking thing about it in the Hebrew Bible is not the emotion behind it but the question of who does the avenging. In most of its occurrences the avenger is God, and the word describes his rightful settling of accounts rather than any private grudge.
When God is the one who takes vengeance, the word points to justice restored: the wronged are vindicated and the guilty are repaid. Whole passages turn on "the day of vengeance," a set time of reckoning, and the famous line "vengeance is mine" (Deuteronomy 32:35) is God reserving that reckoning to himself. It is retribution carried out by the one with the authority and the right to carry it out.
A smaller group of verses puts vengeance in human hands, and here the coloring shifts with the case. It can be a wronged party striking back, as when Samson asks to be repaid for his blinding, or a jealous husband whose fury allows no mercy. It can also be plain malice that the prophets condemn, as with the neighboring nations who take revenge on Judah out of long hatred. The word itself stays the same; what changes is whether the vengeance is just.