I can regret something I did, something I said, something I didn't say (but should have), etc. I can feel pity (sadness, sorrow) for someone's bad luck, their unfortunate situation, that they lost their job, etc. A general rule of thumb to differentiate how/when to use each word might be that regret is about ME whereas pity is about something or someone else.

To feel regret is almost like to feel a deep shame. To feel pity is to feel empathy for someone else. You can pity yourself as well. Examples: I pity her because she has had a lot of bad things happen to her. It is so sad that she died alone. I regret not visiting her while she was still alive.