Right then, apologies for anyone who gets bored.
A card transaction can be processed offline, without contacting the issuing bank or its agent, or online by contacting them to obtain authorisation. Transactions can be done online if the merchant has a phone/data link to their acquiring bank (the bank that processes card transactions for that merchant), and many merchants must make all their transactions online. Certain trusted merchants, such as airlines, supermarkets and TOCs, have authority to perform offline transactions up to a certain amount, for various reasons. These might include speeding up processing or because the merchant cannot go online for technological reasons, such as because it uses rubbishy 11-year old overweight machines with tiny screens
When a bank issues a chip card, it programs several things onto the chip. This includes the number, individual value, and cumulative value of offline transactions that can be performed before the card must go online. (For contactless cards it also includes the number, individual value, and cumulative value of contactless transactions that can be performed before the card must make a chip & PIN transaction.) For some cards, such as the online-only cards mentioned above, these amounts are all set to 0, and if this is the case, a flag is also set on the magnetic stripe of the card which is supposed to prevent the card being used offline in swipe and sign mode. This latter feature does not always work.
All ATM transactions must be performed online, including balance enquiries, so doing a valid ATM transaction will result in the offline limit being reset. Using a card in a normal store won't always do this as the transaction might have been processed offline.