Resignation has two distinct meanings:

1. A formal announcement that you're leaving a job or position. When someone hands in their resignation, they're officially telling their employer they plan to quit. A teacher might submit her resignation to the principal, or a company's president might announce his resignation to the board of directors. The resignation usually includes when the person's last day will be. Sometimes people resign because they found a better opportunity, and sometimes they resign because they disagree with how things are being run or feel they can no longer do the job effectively.
2. A quiet acceptance of something unpleasant that you can't change. When you face a situation with resignation, you've stopped fighting against it and accepted reality, even though you don't like it. Imagine studying hard for weeks but still struggling with a difficult subject. Eventually you might feel resignation that it's going to take longer to master than you hoped. Or picture a soccer team losing 5-0 with only minutes left: the players might show resignation, knowing the game is lost. This kind of resignation isn't giving up on everything. It's recognizing when continued struggle won't change the outcome.