I wrote this document library description in 2007 but now we're on 2013 it doesn't seem to appear? Where would the users see this library description?
There should be a tiny (i) like icon to the right of your library title. Click on that, and you should get the description!
Their are couple of ways to see the description of Document Library.
Method # 1 For all sharepoint versions, you can go to document library settings page and you will see the description on that page. this is for all versions of SharePoint.
Method # 2
Go into Library Setting >
Then form the left hand side, under General Settings (top left of the body of the page), select "List name, description and navigation"
You can now edit the "Description" field. Click Save.
Then the new description will show up on the "i" next to the Document Library name as shown from above example with highlight.