Someone asked me why it was necessary to have the apostrophe s ('s) as well as the word 'of' in a structure like the following: a friend of my mother's. 'Why not just "a friend of my mother"?' she asked, since the 'of' already shows possession.
The 18th-century nitpicking grammarians got uptight about this so-called double genitive, and pronounced it incorrect. But their reasons for doing so were mainly because they were in the habit of comparing English to Latin; because Latin did not allow a double negative, nor, so they believed, should English. However, English is not Latin, and this is a very old construction in English. It had existed long before the 18th-century nitpickers started worrying about it.
In the sentence above, you could remove the 's from 'mother's' and still get a clear, unambiguous sentence, which would also be correct English: a friend of my mother. However, if you change the word 'friend' to 'picture', ambiguity is an issue. 'A picture of my mother' is a picture depicting my mother; 'a picture of my mother's' is a picture that belongs to my mother - quite different meanings.
This construction is called the double genitive, double possessive, or post-genitive. Note that if you are using a pronoun instead of a noun, the possessive is the only possibility: a friend of yours, an uncle of mine (not 'you' or 'me').
The apostrophe is used for other purposes apart from indicating possession. It is used in phrases like 'an hour's time', 'it's all in a day's work'.