Spider's Silk

Spider's Silk is dropped by low-level spiders, from level 16 to 36. It is used as a component in recipes from several professions. Its rarity combined with the value of the resulting crafted items make it a valuable drop easily sold on the auction house.

Source
This item drops from spiders with levels 19-24, and very rarely from other spiders. But with drop rates in the neighborhood of 5%, farming this item will be tedious at best. This is reflected in the AH price which can be 100 times the price of the next comparable item, Thick Spider's Silk. In terms of raw drop rate, the best option is Zarakh on Bloodmyst Isle, but Zarakh has a 5 min. respawn timer so you're probably better off sacrificing drop rate for numbers.

For level 20-30 players, a good place to farm is Thelgen Rock, going for the Cave Stalkers, in south-central Wetlands. Other possibilities include:
 * Hillsbrad Foothills
 * Battle Creeper
 * Forest Creeper
 * Domesticated Mine Creeper
 * Domesticated Creeper
 * Ashenvale
 * Wildthorn Venomspitter
 * Wildthorn Stalker
 * Wildthorn Lurker
 * Duskwood
 * Venom Web Spider
 * Black Widow
 * Black Widow Hatchling
 * Wetlands
 * Leech Stalker

Leatherworking

 * Toughened Leather Gloves (level 135)

Tailoring

 * Lesser Wizard's Robe (level 135)
 * Robes of Arcana (level 150) (used in a Warlock quest to create the Enchanted Gold Bloodrobe.)
 * Spider Silk Slippers (level 140)
 * Spidersilk Boots (level 125)
 * Spidersilk Drape (level 125)