When you sacrifice something or give it up, that's relinquishment. The relinquishment of your position as student council president at the end of the school year might make you a little sad.
Use the noun relinquishment when you're talking about the act of handing something over or abandoning it. The relinquishment of worldly goods is important is some religions, while in others the important relinquishment is that of unkindness. Relinquishment comes from the verb relinquish, to give up. The Latin root of both words is relinquere, "leave behind, forsake, or abandon," which combines re, "back," with linquere, "to leave."