The appraiser skill relates to trading and item value estimation. In order to see your fortress' monetary value estimates in the z-status menu, as well as any individual item values from your stockpile records, your broker must be at least a Novice (level 1) appraiser. (Note that this is not necessary for the display of item quality, which can always be seen via the indicators bracketing the item names.)
Your fortress's current representative trader gains experience in the appraiser skill the first time the trading screen is opened for each caravan visit, but the item valuations in the trade depot are determined by your broker's appraiser skill, not your fortress representative's. (You can, however, reassign the position of broker to your fortress representative to display item values at the depot.) Appraiser has no associated assignable labors.
A good way to train another dwarf is to assign them as broker, make them go to the trade depot and trade, then exit and assign the "main" broker again and make them do the actual trading. This is a very effective way of training the skill, as simply opening the trade interface will instantly turn a completely inexperienced dwarf into a Skilled appraiser (level 4), in the case of dwarf merchants. Don't expect the same gain when trading with elves, as they bring far fewer goods.
(Note: Each of your broker's successes/failures is recorded in the world log and on the dwarf thoughts and preferences screen, and can be reflected in any later engravings in your fortress. To avoid this, use a special burrow for your depot and broker to isolate it.)