How does credit card processing work?
Answer by Daniel Marchant:
Just to add to's answer.
The payment processors actually do a lot of the white label card issuing as well since the card numbers usually originate from them as well as the embossing of the cards. The bin number (6 digits on the front of your card number) represents the sponsoring bank that backs the card.
The payment processor usually is the one linking all the backend pieces together for the banks that use them. Visa and Mastercard act as a pass-through to the processor that actually manages the transaction records for the accounts and settles with the banks based on that above bin numbers. There is quite a bit of accounting on the payment processor side required to keep money flow in check.