To regret something is to wish that it didn't happen, and to repent is to admit an error and promise not to make the same error again. The word 'repent' typically has a religious connotation; to repent something in religion is usually to admit a sin and promise to lead a purer life in the future. 'Regret' has a more common use, and can be used to describe a menial situation, such as regretting leaving an umbrella at home, or a more serious situation, such as regretting drinking and driving.
