All cruise lines operate a cashless system for any purchases made on-board, or even on their private island locations, such as Disney's Castaway Cay.
You will often be required to register a payment card (credit or debit) in advance of leaving for your cruise as part of the Online Check-In process. Some cruise lines will allow you to open a Cash account when you arrive at the port. Normally you will need to present the Payment Card you have registered upon arrival at the port for embarkation with the cruise line Check-In staff.
Every time you make a purchase on-board you hand over the Keycard issued to you by the ship and the purchase is added to your account. Some cruise lines don't use Keycards anymore and have alternative devices, such as Princess Cruises 'Ocean Medallion'.
You can often keep track of the spending on your on-board account via the interactive TV in your room. Just prior to the end of the cruise you will be presented with an itemised bill. Any concerns about the items listed should be raised with the Purser's Office or Guest Relations on-board.
The bill will be settled against the Payment Card you registered during Online Check-In. If you have a card issued in the UK using Pounds Sterling but your cruise line operates in a different currency on-board, you may be charged a transaction fee for the currency exchange.