Yuo need a third party to handle the payments.
PayPal and NoChex are two that are qute well suited to smaller websites.
'Proper' credit card handling requires quite a lot of money to set up by comparison. You need a special type of merchant bank account, an account with someone like Verisign, sometimes a secure certificate for your web server and all kinds of other things.
I'd start with the PayPal click here and NoChex sites click here for lots of very useful information.