Royal Caribbean International is a cruise line based out of Miami Florida, but founded in Norway. They are the largest cruise line by revenue and second largest by passenger count with Carnival Corporation leading them. Royal Caribbean operates 26 ships including several of the largest cruise ships in the world. They operate under the parent company of Royal Caribbean Cruises Ltd which also fully owns Celebrity Cruises and Azamara Club Cruises.
The MyCruise Rewards Program is only accessible to those who hold the Royal Caribbean Visa credit card or the Celebrity Cruises Visa Signature Card. You will not be able to earn points in this program otherwise. MyCruise Points are not to be confused with Cruise Points which are used to calculate your elite status within the Royal Caribbean Crown & Anchor Society loyalty program.
Royal Caribbean and Celebrity Cruises both offer one credit card that earns MyCruise Rewards Points. Both cards are pretty much identical in terms of rewards and perks, so I would recommend choosing based on which cruise line you use the most.