My and mine are both possesive words, so the terms "my friend" and "a friend of mine" mean the same thing. My friend is more likely to be used when the person is with you. So introducing someone, especially to people you know, you are more likely to say "I would like you to meet my friend Zahra." When the person is not with you and someone asks "Do you know Zahra?", you are more likely to say "Yes, Zahra is a friend of mine.". However, both statements are appropriate in any circumstance.