You have analytical curiosity. ^0^ Let's look into the original meaning of 천만에요! 천만(10,000,000) means 'ten million'. It's a enormous number to ancient people. When someome is asked, he ansers '천만에요', it means that there are no possibility or probability. The first meaning of 천만에요 is no posibility. You can use 천만에요 as strong denial. * Oh, (dear) no!; Nothing of the kind!; Far from it.; Never happen.! ; By Jove!; (Stuff and) Nonsense!/ How absurd!/ Don´t talk such nonsense. Secondly, it is used in humble speech. * Don´t mention it, Not at all, Never mind, You´re welcome A: 와 줘서 고마워요 Thank you for coming (A's meaning is containing that you may be busy and spend money to come to me. But you have come. So I thank you for your coming in spite of the difficulties.) B: 천만에요! Don't mention it! (= 고맙긴...! <- 고맙기는 뭐가 고마워! =You don't need to say so! 친한 친구사이에 하는 말) (B is saying that there are no probability to meet dificulties for himself. Actually he expresses that he doesn't care about A's worry.) As you know, 천만 is a noun. But if you say '천만이다', it means 'it's 10 million.' When you use 천만~ as the meaning of 'you're welcome' or 'nonesense', you have to say '천만에(요), 천만의 말씀...' So you'd better understand that 천만에요 is an abbrivation of '천만의 말씀'. (천만의 말씀: You are saying a word that never happen) '미안하다' -> 미안해. 미안해요. 미안합니다 / an adjective '천만에...' -> 천만에. 천만에요. 천만의 말씀.(these are the mostly use expression) / a noun+ a postpositional word Wow... it's difficult for me to explain it in English.

"I'm sorry" is = 미안하다, 미안해요 미안하다 is base form (talk down). 미안해요 is honorific. case by case.