To "avenge" or "take revenge" or "execute vengeance" is to punish someone in order to pay him back for the harm he did. The act of avenging or taking revenge is "vengeance."
* When referring to human beings, to "take revenge" could be translated as "pay back" or "hurt in order to punish" or "get back at."
* Depending on the context, "vengeance" could be translated as "punishment" or "punishment of sin" or "payment for wrongs done." If a word meaning "retaliation" is used, this would apply to human beings only.
* When God says, "take my vengeance," this could be translated by "punish them for wrongs done against me" or "cause bad things to happen because they have sinned against me."