I personally use "a friend of mine/hers/his" when talking about the person to somebody else. E.g. "A friend of mine told me that they are going to Canada". You can also use "my friend" in this context, e.g. "my friend told me they are going to Canada. They are both ok for this situation. However when addressing the friend directly it is most natural to use "my friend" rather than "friend of mine". E.g. " you are my friend", "you are my student". Using "you are a friend/student of mine" would make sense, but would be unnatural. Hope this helps a bit.

Mine is a pronoun, and my is an adjective. Use "mine" as a noun when the object it is replacing is clear and you are saying that it belongs to you. For example, "We all have cars, but mine is the fastest." In this case, "mine" replaces "my car". Use "my" in front of the object you're describing.