A card network is the system that carries every card payment between the shop's bank and your bank. Visa, Mastercard and RuPay are the card networks that most Indians use. RuPay is India's own card network. It is run by NPCI, the organisation that also runs UPI.
A card network does not give you your card, and it does not sign up shops. Your bank gives you the card. This bank is called the issuing bank, or issuer. The shop's bank signs up the shop. This bank is called the acquiring bank, or acquirer. The card network connects thousands of issuers with thousands of acquirers.
Take, for example, a ₹2,400 dinner bill at a restaurant in Hyderabad, paid with a RuPay debit card. The restaurant's acquiring bank sends the payment request to RuPay. RuPay reads the first digits of the card number. These digits tell it which bank issued the card. It sends the request to that bank's card issuing platform. That bank approves or declines the payment, and RuPay carries the answer back to the restaurant. All of this takes about two seconds.
The card network does three more jobs. First, it gives each payment a fraud score, a number for how likely the payment is to be fraud. Second, it creates tokens. A token is a different number that represents your card. So your real card number is not stored on your phone or with the shop. Third, it runs settlement. At the end of each day, the network works out how much each bank owes every other bank. The money then moves between the banks. This is the second half of authorisation and settlement.
Card networks now compete with UPI, which moves money from one bank account to another with no card at all. So the card networks are also building account-to-account payments of their own.
Card networks also carry payments outside shops. For instance, when you tap a bank card at a metro gate, the fare travels over the card network. This is called open-loop ticketing.