I live in Taiwan.There are occasions where I need to thank people, and sometimes they use "不會" to reply my "謝謝". Is this correct and what does "不會" in this occasion mean?
When to reply to "thanks/thank you",
In China mainland, you can say
In Hongkong, you can say
Only in Taiwan, you say
（all expressions used in mainland are also understandable in Taiwan)
Remind: If you say 不会 in China Mainland when replying to "thanks / thankyou", people get confused. (Imagine when you say "thank you", the one you thanked say " No I can't " back to you.)
I'm from Taiwan. 不會 in this context essentially means "No trouble at all", "I didn't really do much".
It's just another humble/polite expression we like to use. And saying 不會 is quicker and easier to pronounce than 不客氣 in my opinion.
When I first(?!) heard the expression/usage form my classmate in high school, I felt that it was very polite and sophisticated. I can understand the implication even if I didn't use that expression myself before that.
You might stumble on this term because it literally reads "not able". But in Hokkien, we can just use this one-syllable word 袂 (see dictionary). It's a negation word that simply means "not".
When it's translated into Mandarin, we somehow have the tendency to make it a multiple-syllable word 不會.
Furthermore, when our buddies say "thank you", we sometimes reply with "三八/三八兄弟" in Hokkien.
(where 三八 means silly/foolish and 兄弟 means brothers/friends )
This 三八兄弟 expression essentially means "Silly you, you don't have to mention that at all." And "We are family, family help each other."
Some of us don't like to mention what we did for others or make people think we want favors in return someday. I'm not sure it's a cultural thing (social graces category, no? ) or a personal thing. I said that because some fellows are REALLY into calling in favors. And personally if I did something that was very difficult to do for others, I do expect appreciations secretly.
So do we really mean "No trouble AT ALL" when we say 不會?
Well. it depends. Normally we do, you can sense that from the micro facial expression and the whole situation. But oftentimes it's just a social expression or a quicker way to say 不客氣.
It is okay to say "不会" in some context. For example, sometimes people help you with something, you fell uneasy to bother and grateful with the help, you would say "谢谢", it will imply that you are actually saying "麻烦了". (Thank you. And I was sorry having bothered you.) People would say "不会".(That's okay. It had not bothered me.)
In terms of whether it is "correct" or not, I think we cannot say it is correct or not, but that it is the idiomatic way or customary way in Taiwan, so it is "correct" or "appropriate" to say that.
For example, in Hong Kong or Guangzhou, when a waiter or waitress brings you a pot of tea in a Dim Sum restaurant, we usually say "唔該", which literally means "Not Should", and it means, "You shouldn't". But the waiter or waitress really should bring customers a pot of tea, so why do most people say "You shouldn't" or "唔該"? Because it is the customary way of saying it, and it gives the meaning of "You are so kind; you really shouldn't (be so kind and go through the trouble of bringing me some tea)."
And besides the page 为什么台湾人回答谢谢的时候喜欢说不会，请问是不会什么 as suggested by @songyuanyao (which I can't say whether it is true or not, as I don't live in Taiwan and don't speak the Taiwanese language yet), but sometimes I feel when they say "不會", it is like this:
When a person says: "謝謝", if his tone is very sincere and expresses a lot of gratitude, it can imply "Thank you so much, for offering me this help, and I appreciate it." And the "不會" literally means "It won't be" or "it wasn't", but it gives the meaning of "No, it wasn't such a big help; you really don't have to thank me." This is how I feel of it, and languages, if 99% of people say it that way or intend to give the meaning by those words, it really becomes the standard way of expressing something, just like when Cantonese people say "Not should" or "You shouldn't", it really means "thanks for your help."