If you relinquish something, you let it go. You relinquish control of the army when you resign as general. You relinquish your plan to sneak into town when your parents find out what's going on.
Relinquish is also commonly used to mean physically letting go of something: The monkey wouldn't relinquish its grasp on the banana. Relinquish descends from Latin relinquere, from the prefix re- "again" plus linquere "to leave."