Regression tests are usually end-to-end tests that should prevent you from making the same mistake twice. When you (or your QA team or customers) discover a bug in a production system, it is not sufficient to apply a hotfix and forget all about it.
One of the things you need to do is write a regression test that should prevent the same error from ever entering the production system again. Good regression tests can even prevent the same class of errors from entering production. After all, once you know what you did wrong, you can imagine other ways to mess things up. Another thing you can do is perform root cause analysis.
