Someone seeking revenge is looking to retaliate for a wrong that has been done.
Sometimes a word can be used as either a noun or a verb, without changing the spelling. That's the case with revenge. You can take revenge (noun) on someone who hurt you, or you can revenge (verb) the hurt, punishing the person who wronged you. While revenge may be sweet, it usually just leads to more revenge from the other side, and so on. While that makes a good Shakespeare play, it can get pretty wearing in real life.