A lot of these words have multiple meanings, so these terms can change depending on where you use them. Generally: 1) Minority group: Any group of people who are not of a majority. The distinction is not fixed- can be due to race, religion, political views, educational level etc. 2) Ethnic minority: This is a bit more specific to a race. So long there is a larger recognised racial group, then anyone else would be in the ethnic minority. 3) Ethnic group: This is also specific to a race, but does not describe how many (or if they are in a majority or minority group). It just means what race you are.

Darryl has given you an excellent answer. I just want to focus on your question. Your question should be 'What is the difference BETWEEN these words?'. Your use of 'among' is a classic example of a linguistic error - a wrong usage which the speaker believes to be correct. The false idea that "We use 'between' for two things and 'among' for more than two" is one of the commonest pieces of misinformation about the English language around today. In fact, we can use 'between' for any number of items, providing we are considering them individually, as we are here. We can only use 'among' when we are not considering the items individually, as in 'There is widespread misunderstanding about the use of these words among learners of English.' I hope that's helpful.