1. If you give up something (or give something up) that is bad for you (for example alcohol, smoking, and eating fatty foods) you stop doing it or having it.
Example: Eric gave up smoking two years ago.
2. To give up something (or give something up) is to stop doing a job, or something else you do regularly.
Example: He gave up work to look after his children.
3. To give up something (or give something up) is also to stop doing something because it is too difficult for you to continue.
Example: He wanted to finish the marathon but he had to give up after ten miles.
4. To give up is to stop trying to think of the answer to a question or problem, or a joke.
Example: I give up. I don’t know the answer. Tell me what it is.