Typically you'd send the required information to an online payment processing service using some kind of messaging (encrypted of course!).
What protocol and message format to use depends on the service you contract with.
This can be a bank or a specialised company.
After validation of the creditrating of the customer they will typically send you a message that payment is OK after which you can notify the customer that his order is being processed.
How it goes after that depends on your contract and business processes.
Either you notify the service when you're shipping the goods to do the actual billing or the customer gets billed immediately and you get notification that payment has been made so you can ship.
Again based on the contract etc. you will then be paid by the payment service after each order or (usually cheaper for you) after a set period (say when total amount paid to you reaches X or once a month).
Indeed contact your bank or a creditcard agency for procedures to follow. I've not been involved in selecting or implementing such services but have done some study as to their operation
