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.