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.
High proficiency in the tanning skill only affects how quickly the raw skin is processed, as leather does not have quality modifiers. However, when a large number of skins needs to be processed, perhaps after butchering a large quantity of livestock, time is of the essence. This can also be solved by having multiple tanneries and tanners available.