Usually 2/3 formal nights, then a mixture casual and smart casual (meaning long sleeved shirt and perhaps even a jacket).
Make sure that you sign up for the meet&mingle party that they throw. You go to royalcaribbean.co.uk >contact us> FAQs > and then under the On Board section it tells you how. You will have to register (it is free). It is a private party, the Captain attended our last one, and it is an excellant way to meet and converse with your fellow travellers before hand.
Smart casual on most nights (IE wear what you like except jeans) Bow tie and trimmings if you like if not smart suit and tie. Ladies can dress how they please (except jeans)