General "royal":
* королевский (korolevsky) - This is the most common and general word for "royal". It comes from "король" (korol), meaning "king".
* царский (tsarsky) - This is also used for "royal" but it specifically refers to the Tsar (Emperor) and his family.
Other words:
* монарший (monarshiy) - This is less common but means "monarchical" or "relating to a monarch".
* дворцовый (dvortsovy) - This literally means "palace" but can be used to refer to something associated with the royal court.
Examples:
* королевский дворец (korolevsky dvorets) - royal palace
* царская семья (tsarskaya semya) - royal family
* монарший трон (monarshiy tron) - monarch's throne
* дворцовый бал (dvortsovy bal) - royal ball
The best word to use will depend on the specific situation and the nuance you want to convey.