You should get a copy of the BPA pedigree certificate. It's just a single sheet of paper. Once the transfer of ownership has taken place (the seller will do this), you will get a fresh certificate from the BPA with your name on it as the new owner. Just like buying a car (except you only get the previous owner's name on it, not all the previous owners).
If you are planning to breed pedigree litters in the future, you will need to join the BPA (unless your pig is a British Lop or a KuneKune - those breeds manage their own herd books). Unless you are a member, you will not be able to birth notify or register any litters.
There is some useful info somewhere on the BPA site under the "pedigree pig" section which explains everything for new owners - but do continue to ask questions here on the forum or PM/email me for info. If I can help, I will.
Good luck
Liz