die Berufung
appeal
A legal remedy by which a judgment is submitted to the next higher court for review.
Der Anwalt hat gegen das Urteil sofort Berufung eingelegt.
The lawyer immediately filed an appeal against the judgment.

appeal / calling / appointment
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appeal
A legal remedy by which a judgment is submitted to the next higher court for review.
Der Anwalt hat gegen das Urteil sofort Berufung eingelegt.
The lawyer immediately filed an appeal against the judgment.

'Die Berufung' (the appeal/calling/appointment) is derived from the verb 'berufen', which means 'to appoint someone to an office' or 'to refer to something'. The ending '-ung' turns the verb into a feminine noun.
Er wurde auf den neuen Lehrstuhl berufen.
He was appointed to the new chair.
You can easily recognize the meaning of 'Berufung' (appeal/calling/appointment) from context. Words like 'Gericht' (court) or 'Urteil' (judgment) point to the legal meaning. An academic context with 'Universität' (university) or 'Professor' (professor) refers to the appointment. A personal context about life goals points to the meaning of 'Lebensaufgabe' (life’s work).
Der Professor nahm die Berufung nach München an.
The professor accepted the appointment to Munich.
In a legal context, the fixed expression 'Berufung einlegen' (to file an appeal) is central. You file an appeal 'gegen' (against) a judgment.
Die Verteidigung wird gegen das Urteil Berufung einlegen.
The defense will file an appeal against the judgment.
When it is about one’s life’s work, 'Berufung' (calling) is often combined with verbs like 'finden' (to find), 'spüren' (to feel), or 'folgen' (to follow). You can also 'als seine Berufung ansehen' (regard something as your calling).
Er hat in der Arbeit mit Kindern seine wahre Berufung gefunden.
He found his true calling in working with children.
In an academic context, you 'erhält' (receive) or 'bekommt' (get) a 'Berufung' (appointment). The destination of the appointment is given with the preposition 'an' and the accusative.
Die Forscherin hat eine Berufung an die Universität Oxford erhalten.
The researcher received an appointment at the University of Oxford.
assignment / order / mission
'der Auftrag' and 'die Berufung' can both mean an important task in life. 'Auftrag' emphasizes more a responsibility or mission that someone takes on or that is important to a group. 'Berufung' means a strong inner feeling: someone believes they are meant for a profession or task. The words can be confused because both can sound very personal and idealistic. The difference is: an 'Auftrag' can also come from outside; a 'Berufung' comes mainly from within.

Mia spürt ihre Berufung, seit sie zum ersten Mal in einem Krankenhaus geholfen hat.
Mia has felt her calling ever since she first helped out in a hospital.
Für den Verein ist es ein Auftrag, jungen Menschen Chancen zu geben.
For the association, giving young people opportunities is a mission.