Selecting a task list as your default "Completed" list helps Nifty understand that the tasks in that list are finished. This feature is what allows you to Automate Your Milestone Progress.
Note that the default Completed task list in every project is titled "Completed", though it can be renamed and still retain its Completed status or you can create and select another task list as your default list.
You can choose which list is your Completed list with the lightening bolt icon. Your Completed list will have a green accent.
You can move a task between lists (even across projects!) from inside it by either selecting the list you'd like to move it to, or moving it directly to your Completed list by hitting the check button.