Festivals are events that occur during certain times of the year.
Each festival has a corresponding NPC that appears in the Warp Zone and has a quest. Enemies have a chance to drop a festival-specific item, with more powerful enemies having a higher chance. The item is used to complete the festival quest, and as upgrade material for some of the festival-specific items.
Multiple festivals can run at the same time. In this case, enemies may drop each of the applicable festival items.
If you save in the Warp Zone, the location of the festival NPCs is also saved, so if the active festival changes before you next load the game, you will need to exit the Warp Zone and return to update which NPCs are there.
Festival timing relies on the system clock, therefore you can cheat by changing the system clock or using applications such as RunAsDate.