From what I understand, this option is only available in lists where the Assigned To Column is present within the content type.
It is used to notify the person or group that the task has been assigned to them, rather than to notify the creator that someone has picked it up. When your create a new item in a list where this enabled the following message appears on the newForm
“The content of this item will be sent as an e-mail message to the person or group assigned to the item.”
As far as i know, the due date plays no part in it
This feature is present within all versions of SharePoint 2010, including foundation. And is Not present at all within SharePoint 2013 as far I am aware(I guess it wasn’t popular).
There is some infomation about it being used in SP2007 here
Edit: Some further reading in this blog post