Traditionally you used to either swipe a card in the store, key the card in a Zon device or you used payment processing software like ICVerify for physical or mail order processing. Internet payment processing involves transaction processing over an open Internet connection, so the transaction can be performed online, potentially even on the Web server itself. Internet transaction services typically provide an API, an HTML type input terminal for manual entry and full online reports of transactions and batches. Transaction processors are separate from the banks that provide a merchant account, although some providers provide a single transaction statement seemingly providing a single service. Once the logistics are set up and you can actually take credit cards and process them over the Web, you need to integrate the software into your application.
Most providers have an API that allows code access to processing functions.
API architecture also varies from client server pieces and pure server pieces. Client server pieces are more difficult to set up as the client side needs to be configured with secure keys and requires the server to be configured to match. Pure server side APIs typically require no client setup and are more flexible as you can use HTTP tools of your choice. Your merchant transactions and Internet transaction fees tend to be charged separately. Be very careful and shop around.