The statement, "I give up," is an announcement that you make when you want to quit doing something.
When you "give up", it means that you quit a game and accept that you lost. This can be used for other situations aside from games as well. For example, you can use it to describe debates:
We argued for a while about what restaurant to go to, but I finally just gave up and let him decide.
"Give up" is similar to "quit", but giving up is more about your feelings and motivation, while quitting is an official action that you announce to people.