Paying bills is usually done by wiring money using the bank's online banking interface (or paper) or by automatic billing through the bank account. Credit cards aren't used to pay bills; paying in stores is usually done with debit cards (Maestro) or credit cards (but people rarely use them).
The online system
is used only to purchase over the web; it's way more popular than credit cards (everybody has a bank account and fees are low). The API system works a bit like PayPal.