Generally, this is a method that allows the sender of some article to know that their delivery was made, or not.
Specifically, the USPS will add a sticker (PS Form 152) with a serial number and bar code to any delivered letter or parcel, information about which can be accessed online, by phone (1-800-222-1811) or by visiting the post office. The date, time and zip code of delivery are all available with this option. You can also find whether the delivery was made, failed, or forwarded. The retail price for Delivery Confirmation is $.45 for Priority Mail and $.55 for all other options.
Apparently, it is fairly common for the article to be delivered and the delivery information to not be entered into the USPS database. In such cases, a refund for the fee can be requested using a PS Form 3533. (But I can’t imagine the hassle to be worth the fifty cents!)