The unsubscribe link sends the person who clicked it off to our software, where the click is recorded, and the software works out which subscriber, list and campaign it is associated with.
That person is immediately changed from active subscriber to unsubscribed in that list, no need for you to manually do anything at all. In the campaign report, that subscriber will now appear as having unsubscribed.
Suppression list, and unsubscribe settings
By default, unsubscribing from a list will only unsubscribe that address from that specific list. So if the address is on your 'customers' list and your 'newsletter' list, they will still be an active subscriber on your other list.
You can however configure all unsubscribe requests to be added to your suppression list automatically. That will in turn remove any subscriber that unsubscribes from all of your lists and it will keep them off all of your future lists.
To change the default behavior, so that all of your unsubscribers are added to your suppression list, click on "Subscribers" and then "Unsubscribe Form" and select "Automate Global Suppression List".