This might sound like a very basic question, but what is the difference between the words nasılsınız and nasılsın? I know that they mean the same thing. 

When I was taught Turkish, my teacher said that the  ´siniz´ form is more polite, and used if the person is not a close relative or friend, or you are talking to a group (plural form). It is often used if the person is older than you, to be more polite/respectful.

If you are addressing someone you know well, a friend or a relative then the shorter ´sin´ suffix is used, but no-one would take offence if you used ´siniz´.

This is also true for using the pronouns sen & siz