If you want to accept credit/debit card payments online, you're going to need two things:
- an Internet Merchant Account (a business bank account which can accept such payments; there's usually quite a bit of business compliance, risk and anti-fraud attached, but if you're a serious business with an existing business account, talk to your bank)
- a Payment Services Provider (who provide you with, for example, an API or a hosted checkout solution, and who act as the go-between for your web site and your IMA, which will typically only provide very limited options)
There is, of course, a fast growing trend of alternative payment services; PayPal, Envoy, MoneyBookers, Neteller, uKash, etc. Those are definitely worth exploring, because they tend to provide a certain degree of risk-free/guaranteed payment, and they dramatically widen the numbers of customers you can reach - think in terms of people who aren't old enough to have cards, or who aren't able to get them due to debt problems or the like.
Perhaps the easiest combination is to find a Payment Services Provider who can handle the whole lot - set you up with an IMA (or rather, underwrite your risk with a bank so they'll be happier to give you one), provide payment processing services, maybe a bit of fraud protection and risk management on top, chargeback management, and then handle all those possible payment methods, including things like 3D Secure (Verified by Visa/Mastercard SecureCode) which can be a real headache to integrate with.
If you're going to be getting into the business of handling online payments, even if it's just an occasional thing, then you'll need to knuckle down and do a bit of serious research into what you need, what you don't need, and what you're willing to pay, 'cause payment service providers obviously charge for this stuff, and it'll be a serious business partnership.
For once, Wikipedia seem to have an appallingly lightweight list, so you may want to Google "payment services provider".