There's a lot of info about the new (not really so new now) tiered suite program, but it really only applies the Oasis class and up class of ships.
http://www.royalcaribbean.com/royalsuiteclass
If you're sailing on smaller ships, it varies slightly from ship to ship but is a pretty standard set of perks for all suites (Grand Suite and above). This typically includes:
Priority embarkation
Priority tendering
Concierge lounge with daily cocktail hour
Priority reservations for specialty and my time dining (and spa appointments)
MDR menu available for room service
Suite breakfast (for suite guests only in specialty restaurant)
Special sun deck and seating for shows