deputy / representative
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deputy
A person who is officially assigned to represent another person (usually a superior) in their absence and take over their duties.
Während der Direktor im Urlaub ist, leitet sein Stellvertreter die Abteilung.
While the director is on vacation, his deputy runs the department.
The word is a compound made up of 'Stelle' (position/place) and 'vertreten' (to represent/stand in for). A 'Stellvertreter' (deputy/representative) is therefore literally a person who represents the place of another person.
Mein Chef ist krank, sein Stellvertreter übernimmt heute die Leitung.
My boss is sick; his deputy is taking over management today.
To say whom a person represents, the genitive or the preposition 'von' (of/from) is usually used. The genitive construction is more common in written language.
Frau Meier ist die Stellvertreterin des Abteilungsleiters.
Ms. Meier is the deputy of the department head.
Common verbs in connection with this role are 'zum Stellvertreter ernennen' (to appoint someone deputy), 'als Stellvertreter fungieren' (to act as deputy), or 'jemanden als Stellvertreter einsetzen' (to assign someone as deputy).
Er wurde zum Stellvertreter des Bürgermeisters gewählt.
He was elected deputy mayor.
You can recognize the abstract meaning by the fact that the person does not stand in for another person, but serves as a symbol 'für' (for) a group or an idea. This usage is often found in literary or sociological texts.
Die Figur im Film ist ein Stellvertreter für die Jugend ohne Hoffnung.
The character in the film is a representative of youth without hope.
representative / substitute
'der Vertreter' and 'der Stellvertreter' can both mean a person who acts for another person. However, 'Vertreter' is broader: it can be someone in sales, someone with an assignment, or someone who temporarily replaces another person. 'Stellvertreter' is more specific and usually more official. It means a person who steps in for a particular other person, especially when that person is absent. Learners may confuse the words because both have to do with 'acting for someone'. When referring to a fixed substitute for a leadership role or position, 'Stellvertreter' sounds more natural.
Wenn die Chefin im Urlaub ist, unterschreibt ihr Stellvertreter die wichtigen Briefe.
When the boss is on vacation, her deputy signs the important letters.
Der Vertreter der Firma erklärt dem Kunden die neuen Geräte.
The company representative explains the new devices to the customer.