They will deliver one towel per passenger to your cabin on the first day for use both at the pool and ashore. If you lose it, they charge you around $20 for the towel. I am not sure that they have ever charged anyone, but that it is more of a motivator to bring the towels back from going ashore. If you use a towel, your room stewerd will replace it with a fresh one each evening.
I have noticed that towels are made available on at-sea days by the pool, but everyone has been trained to take theirs when they vacate a lounge chair, which frees up all the lounge chairs where people would normally leave the towels for someone else to pick up. The towels are of reasonable quality.
You can see the reasoning behind the move when you consider how many towels must be left in ports on the beach after each visit. It must be very expensive to keep replacing the ship's stock with new towels.
Royal Caribbean does not have self-service laundry facilities, so you would have to have the ship launder any towels that you bring with you. I am not sure what the charge would be, but it is similar to what you would pay to have them laundered at home. It would take two days to get them back unless you did the "express 1-day service" that adds an additional 50% to the cost.
Hope that helps. Have a great cruise.