The Royal Party is held once per month. To join it, be sure you are at level 16+, and your Party Table is placed on your kingdom, and built completely.
After that you can access to Royal Party by clicking the Royal Party Table on your kingdom, or clicking the Royal Party icon in the activity center.

How to play:
Click on the “make” button to make party dishes. You will get 1 Chef's Hat for every finished party dish.
Once you have made all dishes, you can invite friends to your party by clicking the “invite” button. If they accept your invitations, they will give you Bonus Gifts at the end of the party of that round! (You can spend 2 rubies to invite NPCs to your party immediately.)
Wait until the party is over (each party takes 6 hours).
When you have gained enough hats, don't forget that you can redeem them for one random golden reward.
When the party is over, you will see the finished “Party Is Over” window. On that panel, if there is less than three days until the next party, you will see a countdown timer for the next party. If the next party is more than three days away, you will see “Coming Soon!” instead.
 
 
Q&A
Q: Will my Chef’s Hats be saved for the next party?
A: They will be saved for the next party. Don’t worry. :)
 
Q: What's the benefit I can get from upgrading my party table?
A:  Upgrading your party table enables you to get more space for invitees, which means you can get more chef's hats in one round.