In the Hamlet DLC, Silk drops both from killed Spider Monkeys (at a 25% rate) and from chopped-down Cocooned Trees: 1 piece for a small tree, and 2 for a medium or large tree.
Also, Webber can harvest silk by shaving his beard. In the Reign of Giants DLC, Silk can be harvested from Tumbleweeds. Spider Queens drop 4 Silk (and a Spiderhat) when killed.
A large den yields 6 pieces of Silk, a medium den yields 4 pieces, and a small den 2. Silk is also dropped in bulk when a Spider Den is destroyed by any means except burning, in which case all its drops turn to Ashes. Silk is typically gathered by killing any type of Spider. It can be used to make many useful items like the Sanity-restoring Top Hat, the Boomerang, the Winter Hat for moderate warmth, the Fishing Rod, and Bird Traps, among other things.
Silk is a resource mainly obtained from Spiders.