If you buy a NEW annual pass, the year active period starts the first time you ACTIVATE it at a ticket window or at Guest Relations at WDW (or, if you buy it there, when you buy it, but with the FP+ most guests will probably buy theirs ahead of time). Not the first time you use it to get into a park. You can't go into a park with it until you activate it, and once you have activated it you do not have to use it to enter a park (although most do). If I were to buy an annual pass today, April 28, online or by phone, and go to WDW on June 1 and activate it then, that pass will start running on June 1, 2016 and expire on June 1, 2017.
If you RENEW an annual pass, no matter when you renew or when you activate your renewal, the dates of the active period remain the same as the previous pass. For example, I have an AP that expires December 8, 2016. If I renew it, no matter when I pay for the renewal, and no matter when I return to WDW to activate my renewed pass, the active period of that pass will start December 8, 2016 and will run through December 8, 2017. If I'm going to be at WDW around December 8, it's a good idea to do the renewal. what I would probably do is renew it 60 days before the 8th so I can make fastpasses, and then I would need to activate it at WDW. If I'm going to be at WDW on or around December 8, I can, if I want to, just purchase my renewal there and I don't have to activate anything.
However, If I renew my pass by phone or online and pay for it, but I don't actually go to WDW until April 28, 2017 and activate it then, the expiration date will still be December 8, 2017. At that point, it would probably have been best to let the renewal discount go and activate a new pass on April 28, 2017, which would expire on April 28, 2018.