To "give up" means to quit something. "Giving up" is usually a bad thing, so we often tell people not to do it:
Don't give up! Keep going.
If someone "gives up", they seem weak and undisciplined.
You can also use the phrase "give up on (something)":
What made you give up on your art career?
One way to use "give up" that doesn't sound negative is "give up (a bad habit)":
I'm trying to convince him to give up smoking.
I gave up sugar a few months ago.
In these situations, "giving something up" is a good thing.