Feathers are a resource for practitioners of Bowcraft/Fletching and Carpentry. For the former, they are required for the construction of arrows and crossbow bolts. For the latter, they are needed to make both the Raven Helm and the Vulture Helm. Surprisingly, Tailors do not need any feathers to create Feathered Hats.
To obtain feathers, slay a feather-bearing creature (e.g., Bird, Harpy, Phoenix), use a bladed weapon on the corpse. As with most resources, skinned corpses yield double the amount of feathers in Felucca than they do on other facets (though not on Siege Perilous and Mugen shards).