You need a Merchant or Bureau Account with one of the credit card processing organisations (or banks), a secure server facility, and some kind of ordering system - usually provided as an integral part of a Shopping Cart program and frequently called a Payment Gateway.
Bureau Accounts are particularly popular because they provide a 24-7 credit card checking facility without intervention by the website owner, enabling customers worldwide to buy at any time of the day or night and giving the website owner an "always open" sales facility.
Note that shopping carts are normally integrated into your website and provide buyers with the ability to select and order the products of their choice. When they are ready to pay, all the products they have selected are listed in the "shopping basket", along with prices. By clicking the PAY NOW or CHECKOUT button, the buyer is then transferred to the secure server using a process called a Payment Gateway to complete the purchase by credit card in a safe and secure environment.
Some of the well known credit card processing organisations include WorldPay, PayPal, SecPay, Protx and MultiCards.