Simply put, we regret choices we make, because we worry that we should have made other choices. We think we should have done something better, but didn't. We should have chosen a better mate, but didn't. We should have taken that more exciting but risky job, but didn't.
Feeling regret reminds us to think carefully about our decisions and helps us not to make the same mistakes again. Regrets are also how we learn about ourselves, and know what it is we really want. In feeling regret, we have clarity about what outcome and things we truly want for ourselves.