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.