The only correct one is "A good friend of mine". The second one, "A my good friend", makes no sense. You could say, "My good friend", and it would mean the same thing. Here are some examples of when you could use the different sentences: "Andrei is a good friend of mine" or "I'd like you to meet my good friend, Natalia." They both meant the same thing, it just depends how you structure your sentence in which one you use.

The first sentence: "A good friend of mine. " is correct. In the second one, we don't say "A my good friend". Sara's answer is great, but I think your question is " one of my good friends", right? If that's right, I would say," Anna is one of my good friends." It means: you have many friends, but Anna is the GOOD friend who are really close to you.

‘A good friend of mine’ is correct. ‘A my good friend’ is not correct because ‘a’ and ‘my’ have much the same function (both are indicating which good friend is being referred to), so it doesn’t make sense to use them both – in the same way that it doesn’t make sense to say ‘this that pen’ or ‘some another road’. ‘A good friend of mine’ and ‘my good friend’ are very close in meaning but slightly different in what they imply. ‘I’m inviting my good friend Catherine to join us for dinner’ implies nothing about how many good friends the speaker may have; but changing that to ‘I’m inviting Catherine, a good friend of mine, to join us for dinner; implies that the speaker may have several good friends of whom Catherine is one. (Also, there is a difference in construction: in my first example, the phrase ‘my good friend’ is defining; in my second example, the phrase ‘a good friend of mine’ is not defining - and so needs commas.)

both are corect