Earlier quoted context omitted.
Obviously put country first, then whatever the equivalent of a ZIP code is second. In the UK, a post code gets you down to around 15-16 delivery addresses on average. No-one there really _ever_ needs to type an address.
Most British websites work like this website suggests, in that they ask for the postcode first, then give you a dropdown of all the addresses at that postcode. It works great, except my address is for some reason not in whatever databases these websites use. The building number is on the list, but not the individual flats. So I have to put in the postcode, choose something like "My address is not listed", then fill i…
The PAF is about what the Royal Mail calls "Delivery Points" which are places they promise to deliver physical letters to. Inside my building for example every home has a front door, with a letter box, and letters literally get posted into my flat, but down the street there's a building with a rack of boxes set into the wall and post is delivered to your numbered box, and up the street there's a townhouse converted and all the mail just goes in one piles for everybody in that building.
Many delivery points share a postcode, but because they know them all they actually all have unique numbers, and unlike a postcode they're not for humans so they get changed quite frequently as new buildings are constructed or working patterns change, if you examine your post carefully (in the UK†) there are two rows of fluorescent orange dots on the outside, these were printed by mail sorting machines shortly after the mail was received, one is an entirely arbitrary serial number and it designates that piece of mail for a very short period (say a few days) to enable statistical tracking for performance - if #213940202 entered the system in Glasgow on Monday, was in Portsmouth by Tuesday but wasn't delivered to someone's door until Friday the problem ain't in Glasgow. But the other one we're interested in here is the Delivery Point as a numeric code. If you don't have multiple addresses this row of orange dots will be identical on every item you've received for some time yet it's different on someone else's mail.
† This trick was invented in Britain but is used (licensed) in some other countries because it's a good idea, however exactly what is encoded and so what it "means" may vary.