Tanner is the skill associated with the tanning labor. Tanners turn raw skins from butchered animals into usable leather at a tanner's shop. They have to act fast, as raw skins will rot fairly soon after being created. By default this job will automatically be queued when a skin is produced, but this can be deactivated from the orders-workshops menu. If you're still not getting the hide tanned, enter the order "tan a hide" manually in the manager menu (j-m) with a sufficient amount. Each butchering job yields a single hide, and each unrotten hide yields one piece of leather - regardless of whether the animal is a kitten or an elephantBug:3732.
Tanners also make parchment using milk of lime and a raw hide. Parchment is a somewhat difficult-to-make alternative to paper or papyrus. Parchment made from a cow's hide is called vellum, but is otherwise identical.
High proficiency in the tanning skill affects how quickly the raw skin is processed in both cases. Leather does not have quality modifiers, but parchment does.