Newsletter (as by plugin present in core) has the ability: The plugin has to be installed.
Now when you use the menu's (2 available) and put those in site, AND newsletter (make a test mail first)
when running you will see the button (or alike) in the menu itself (not subscription yet > you can subscribe) : Iff you are already a subscribed person to that newsletter there will be instead of subscribe the unsubscribe button.
(Please check your Signup Page preferences, making sure that 'Subscribe to content/mailouts' is set to 'Display' or 'Required'. You should also have one or more 'Newsletter' userclasses set to managed by 'Everyone'.)
This is for the newsletter plugin ! The systems mail letter is different as that is meant as a 1 one system (from site to spec. user)
Adding an unsubscribe if not present (as i am unaware if it is there) might be possible, but would need extensive coding (underlaying) to achieven the unsubscribe result. (i personally do not use it)