Borks papers are notes that Bork wrote and didn't include in the Book of Dragons. This was the only source of information aside from the Book of Dragons until the Dragon Eye was discovered. Hiccup received them on Bork Week in We Are Family, Part 1. The Dragon Riders often look to the papers whenever they need to find something really important.
History
Dragons: Riders of Berk
When Berk was ready for Bork week, Stoick got Gobber to find his great-great grandfather's old notes, so they could give them to Hiccup. Hiccup was honored to be given Bork's dragon notes and learn the information that he hadn't put into the Dragon Manual. However Mildew had managed to place fuels dragon notes - about Night Furies - into one of Bork's notebooks and Hiccup believed that they were real, until he and Toothless fell into the trap that was set by Alvin, his Outcasts and Mildew.
Dragons: Defenders of Berk
When Snotlout's dragon, Hookfang became ill and week, Fishlegs looked through the papers so they could find a cure.