To punish or harm someone in return for a wrong they did to you or someone you care about.
From Old French 'avengier,' which comes from 'a-' (to) and 'venger' (to punish or vindicate), from Latin 'vindicare' (to claim or avenge). The word originally meant to vindicate or justify, but evolved to mean taking retribution. It entered English around the 13th century.
Avenge is the reason we have the word 'vendetta'—both come from the Latin root for vengeance, and it's literally why the Avengers are called the Avengers: they're avenging the death of someone important!
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