If you're unapologetic, you refuse to say you're sorry or to express any regret. You might be unapologetic about missing your horrible cousin's birthday party, for example.
If someone isn't sorry, she's unapologetic: your English teacher is probably an unapologetic grammarian, and you might be unapologetic about your lack of interest in the art of proper grammar. Neither of you is willing to acknowledge any regret for your feelings. While the adjective unapologetic can describe someone who simply refuses to apologize, it also conveys a sense of strength: an unapologetic feminist, for example, is proud to stand by her beliefs.