Weaver is the skill associated with the weaving labor. Weavers use a loom to weave cloth from silk thread (from webs left by cave spiders, giant cave spiders, or phantom spiders), plant fiber thread (processed from rope reeds or pig tails), or yarn (spun from wool of shearable creatures). Unlike most intermediate goods, cloth has a quality value, and dwarves with high skill in weaving create higher-quality cloth. Collecting spider webs also requires and trains the weaving skill.
Note that, because they visit the caverns often to gather web, weavers are particularly susceptible to encountering dangerous wildlife, such as the giant cave spider creating the webs they're currently gathering. Consider assigning physically stronger dwarves for the job and giving them some war dogs or similar, so that they have a bigger chance of fighting off hostile cavern beasts.