When should you use a private/inner class?

Question

To clarify, what I'm asking about is

public class A{
    private/*or public*/ B b;
}

vs.

public class A{
    private/*or public*/ class B{
        ....
    }
}

I can definitely think of some reasons to use one or the other, but what I would really like to see are convincing examples that show that the pros and cons are not just academic.