PBA is a Bluetooth profile (mode) that allows a phone to expose its contact (phone book) info to another authorized device, such as the hands-free function of a car or car accessory.
PBA allows the car (or car accessory) to handle phone book browsing and dialing. The phone book might be displayed on a display on the dash, or projected onto the windshield, for example. PBA also allows voice dialing to be handled by the car or car accessory, instead of the phone.
Not all Bluetooth devices support PBA. For PBA to work, both devices must have Bluetooth, and both must support PBA.