Anything. It doesn't even have to include your characters. Did you read the example in the calendar? Bilbo Baggins barely appears in The Lord of the Rings, but Bilbo's acquisition of the One Ring of it leads to Frodo's quest later. Another plot background story goes even farther back in time: the story of when Gollum, then named Smeagol, first acquired the ring. Other possible background stories: Sauron first forges the One Ring. Isuldur acquires the ring by cutting it from Sauron's hand. Sauron corrupts the wizard Saruman. Wormtongue corrupts Theoden. Aragorn's father, Arathorn, heir of Isuldur, is killed, and Aragon is placed in the care of the elves for safekeeping while Denethor ascends as steward of the throne. All these stories are critical to the story of Frodo's quest, and any one of them would be an appropriate background story that sets up the plot.

Did you try linking your contest entry properly? If it's not linked properly, the judges won't read it.

Regards,
Michelle