Auftrag. A command, instruction, order, to complete a task, job, assignment; to order a product or service; an obligation, a duty. An Auftrag is given by a manager or a customer. The Auftrag indicates that someone will do something for another. An Auftrag can be rejected. They can be legally binding. An employee can assign herself an Aufgabe, but not an Auftrag.
Aufträge (plural of Auftrag) are foundational to any economy. Whether it is involves one colleague answering the email of another or one company building a production site for another, Aufträge are the lifeblood of commercial activitiy.
An Auftrag is at its core a request from a customer. Taking on the Auftrag signals that one will complete it to the best of their ability. The details are set in a purchase order or in a contract.
German companies report time and again that their Auftragsbücher, order books, are full, but that they cannot fulfill all of them due to a shortage of trained personnel, often technicians and engineers. Taking on an Auftrag is no guarantee that one can complete it.
This also means a certain degree of risk for the Auftraggeber (Auftrag giver), the customer, that the supplier will not supply the end product on the agreed upon date, or at the expected level of quality. In many ways it is also unimportant who completes the task. In contrast to an Aufgabe, an Auftrag is impersonal, business-like, unemotional. The relationship is all about the execution of the job. No more, no less.
A self-identification with the task is secondary. Only the final results count. Is the Auftrag completed, rejected or not doable, then it automatically no longer exists.