What does renouncing mean?

To "renounce" means to give something up, usually by declaring that you have given it up. For instance you might renounce your support for the Liberal Party, meaning you are publically stating that you are no longer supporting that party, or that you no longer subscribe to their beliefs.

In card games, it means you have failed to follow suit. In whist, for instance, hearts were led but you have no hearts and so discarded a low club instead.