Avenge is a verb meaning to inflict punishment or penalty for a wrong doing.
Revenge is a verb and a noun meaning the act of taking vengeance for an act of wrong doing.
An easy way to try to remember it is that Avenge means the achievement of justice, whilst revenge means retaliation.
Revenge is often associated with negative connotations(as in making something bad happen to someone for something they did to you) whilst avenge is generally more seen as heroic and making right a wrong that was done.
I hope that helps.