What does the "past due" trigger do?

There's an "order past due" trigger. Is this for invoice billing or if the order has not hit production by the customer's requested date?

It's for billing. So the "order past due by" condition of a trigger is going by the date payment is due based on what you set for "Default Status to Charge Customers" in financial settings. For example if you charge the customer on the day an order moves to "shipped" status, then a trigger that is set to fire on past due by 1 day, then it should send an email the day after a job is set to shipped.

And of course that's just to tell it WHEN to fire - it's up to you to put the right message in the body of the trigger email!