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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.
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 false dragon notes about Night Furies into one of Bork's notebooks, leading Hiccup to believe that they were real, until he and Toothless fell into a trap set by Alvin, his Outcasts, and Mildew.
Dragons: Race to the Edge
Bork's Papers are often mentioned, usually by Fishlegs, when trying to search for information that he can't seem to find in the Dragon Manual. If any info is found, it's combined with any additional info possibly found in the Dragon Manual and/or the Dragon Eye, to create the most thorough collection of facts possible.
When it's seen, it's usually in an array of papers seen on Fishlegs' table in his hut, or on the table in the Edge Clubhouse.