assignment / order / mission
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assignment
A specific task or work assignment that someone carries out for payment or on instruction.
Die Agentur erhielt den Auftrag, eine neue Werbekampagne zu entwickeln.
The agency received the assignment to develop a new advertising campaign.
The word derives from the verb 'auftragen' (to assign/instruct). When you 'jemandem etwas auftragen' (assign someone something), you give them a task. 'Der Auftrag' is therefore the task that you have been given.
Der Chef hat mir aufgetragen, den Bericht bis Freitag fertigzustellen.
The boss assigned me the task of finishing the report by Friday.
'Auftrag' (assignment/order) is often combined with fixed verbs that describe the life cycle of a task: you can 'bekommen' or 'erhalten' an Auftrag (get/receive an assignment), 'annehmen' or 'ablehnen' it (accept or reject it), 'ausführen' it (carry it out), and finally 'erledigen' it (complete it).
Die Firma hat den Auftrag für den Bau der neuen Brücke erhalten.
The company received the assignment to build the new bridge.
The expression 'im Auftrag von jemandem' (on behalf of someone) means that you do something for another person or company. It is often used in formal and business contexts.
Ich rufe im Auftrag von Frau Dr. Meier an, um den Termin zu verschieben.
I am calling on behalf of Dr. Meier to reschedule the appointment.
When 'Auftrag' (mission) describes an idealistic mission, it is often used with verbs such as 'sehen' (to see) or 'haben' (to have). The context is then not commercial, but moral or social.
Ärzte ohne Grenzen sehen ihren Auftrag darin, Menschen in Krisengebieten zu helfen.
Doctors Without Borders sees its mission as helping people in crisis areas.
work
'die Arbeit' and 'der Auftrag' can both refer to something that needs to be done. 'Arbeit' is more general and means the task itself or the activity of doing it. 'Auftrag' emphasizes that someone has given or ordered this task, often a boss, a customer, or a company. So an Auftrag can be work, but not all work is an Auftrag. Learners may confuse the words because both occur in professional contexts and with tasks. When it is important who gave the task, 'Auftrag' fits better.
Die Firma bekommt heute einen neuen Auftrag von einem Kunden.
The company is getting a new order from a customer today.
Für die Präsentation hat das Team noch viel Arbeit vor sich.
The team still has a lot of work ahead of them for the presentation.