What is the difference between this two sentences - 1. An friend of my father 2. A friend of my father's
They are two of the three genitive forms in English.
A friend of my father.
A friend of my father's.
My father's friend.
They are equivalent here, all of them referring to possession, but the second is only used for possession, making them distinct in some other cases:
A picture of my father.
A picture of my father's.
My father's picture.
Here the first would be used only to state that your father was the subject of the picture, the second only to state that your father owned the picture. The third would be ambiguous between those two readings.