Spider Silk is a Loot, obtained by completing the missions The Way of the Timid, The Art of Gastronomy, Summons, and Gripped by Fear, or by stealing it from Ahriman (loot 4). It is required to create the Ayvuir Red, Masamune, Terre Rod, Perseus Bow, and Germinas Boots.
The Spider Silk and Spider Web are special Accessories that increase EX Revenge Duration by 5% and 10%, respectively. The Spider Silk can be traded for an Attractorb, three Moonflow, three Fal'Cie Husks and 3,500 gil, while the Spider Web can be traded for a Spider Silk, three Time Warps, two Tome of Mysteries, and 3,500 gil.
A third accessory, Spider's Bane, is even stronger, increasing EX Revenge Duration by 15%. It costs 3,500 gil to trade and needs a Spider Silk, three Voidshards, and two Tome of Mysteries.