Expressing gratitude and regret

Thank you.

When expressing gratitude in a polite manner, you can say 감사합니다(gam sa ham ni da). It is more polite when accompanied with a slight bow. However, if you use it with people who are close to you such as friends or family members, it will sound too formal.
To give thanks to these people, it is better to use more friendly expression 고마워요(go ma woe yo) instead.

Don't mention it.

To respond politely to 감사합니다. you can say 별말씀을요(byeol mal sseum eul yo). Literally, it means that there is no need for special mention and is used as polite response to a thank you. This expression will sound humble and modest. If you find the pronunciation, you can simply say 아니에요(ah ni ye yo) instead.

I'm sorry.

When politely expressing regret, generally you can say 죄송합니다(jae song ham ni da).
If the other person is older than you or of higher status, you can say it with a slight bow.
In casual relationships such as with school friends or co-workers in a company, you can use 미안합니다(mi an ham ni da) instead.

It's all right.

When responding to an apology, you can say 괜찮아요(gwaen chan ah yo) to express that there is no problem. This expression can be use it to calm down others by saying 괜찮아요?(Are you all right?) with a rising tone at the end. You can also use it to calm down others by saying 괜찮아요(It is all right) with a slightly declining ton at the end.

Excuse me.

When politely asking to be excused or calling or attention from someone, you can say 실례합니다(sil rae ham ni da) which sounds very formal and courteous. In daily life 저...is also often used to attract another person's attraction instead of addressing them directly using their name or title because Korean don't use "you" when calling others.