This entire site is new to me and, to my knowledge, is all COTS. The employee before me added a custom column (somehow) called "select" to a document library. When the user clicks the "select" button next to their chosen document (in this case a set of arrows), the task list web part above the library displays the tasks/workflow items associated. I was asked to recreate this for another library but am completely clueless how this previous employee was able to create it. Help?
They have not added a custom column. They have linked two list web-parts on the page using a connection. The selection column will appear when you do this. You can find examples of how to do this here;
However, in order for there to be tasks associated with a Document Library I suspect there is some underlying workflow that will have been created with SharePoint Designer. This may include tasks like Approve Document or such-like.
If a task is created for a document it stores the document URL under the Related Content field. However I am unsure how you would connect to this field from the Library webpart. If you Edit the page, click on the Document Library web-part and inspect the connection used you should be able to work out what fields they used.