I believe that Bpay is an Aussie bank thing. It is a single bill payment system that works across ALL banks in Australia.
The various companies have to register with BPay to get a BPay number. Once they have that number, they can quote it as a payment method.
All Australian consumers with a Bank Account can then use that BPay number and pay it directly from their bank accounts or credit card over phone or internet.
This payment method ensures that the companies do not have access to our personal banking details; unlike setting up direct debit (either from a bank account or a credit card).
It doesn't allow for automatic payments; you have to go in and authorise a payment to the specific company for the specific amount. Takes 2 days to clear.
And it isn't necessarily free - the banks have the discretion to charge for the service and a very few of them do. At the moment, banks are charging the companies for the fee of handling each transaction. It is a very small charge but given the volume of traffic, it does add up to a significant amount of money.
How do I know? Well, given that I was previously working on a project at a large TelCo that was looking to "encourage" direct debit payments, I found out a lot more about BPay than I would have liked!

Just goes to show that there is no such thing as useless info!
I guess PayPal might be the closest on a global scale.