Vengeance, vindication, and a few related words are derived from a Latin word pertaining to punishment and retribution. These terms and their definitions are listed in this post.
The forebear, the verb vindicare, means “avenge” and “claim,” though for each of its English descendants, the former sense holds almost exclusive sway. (Interestingly, vindicare may in turn stem from vim dicare, meaning “show authority”; the first word, meaning “force,” has been preserved in the first word in the phrase “vim and vigor,” while the second is the source of dictate, diction, and the like.)
The verb avenge suggests righteous retribution (one who does so is an avenger), while the connotation of revenge suggests malicious retaliation. Revenge is more common as a noun than as a verb (though revenger is not employed to describe one who commits an act of revenge), while vengeance is employed as a noun in place of avenge; the adjectival form is vengeful.
In a political context, revanche, adapted from the French verb revenchier, meaning “revenge,” refers to a policy of reasserting status or recovering territory; the practice is revanchism, and a revanchist is an adherent.
Vindication is synonymous with vengeance, but usually it has the sense of “confirm,” “defend,” or “justify,” or “free from blame”; one who has been accused, then exonerated, is vindicated, and one who performs the vindication is a vindicator. The adjectival form is vindicative, which should not be confused with vindictive, which means “vengeful” or “spiteful.”
For the most part, these terms entered the English language by way of French, but an exception is vendetta, adopted from Italian. Originally, it denoted a feud, especially a deadly one between families or clans that involves back-and-forth retaliation. Now, the sense is of a malignant campaign to discredit or harm someone.