If you puchase the AP online, you will receive a voucher that you will redeem at the park for your AP. The countdown begins on the day that you exchanged the voucher for the AP.
If you purchase you AP at the park, the countdown begins with the day that the AP was purchased, regardless of whether you use it that day or not.
If you have a multiday pass that you are upgrading to an AP and that multiday pass has already been used, then the anniversary date of the AP will be one year from the first day that the multiday pass was used. Note that you cannot uprgrade a multiday pass if it has been more than 14 days since it was first used.