auftragen
to apply
To put a substance such as paint, cream, or another layer onto a surface.
Tragen Sie die Sonnencreme großzügig auf, bevor Sie in die Sonne gehen.
Apply the sunscreen generously before you go out into the sun.

to apply / assign
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to apply
To put a substance such as paint, cream, or another layer onto a surface.
Tragen Sie die Sonnencreme großzügig auf, bevor Sie in die Sonne gehen.
Apply the sunscreen generously before you go out into the sun.

'Auftragen' (to apply / assign) is a separable verb. The prefix 'auf' comes at the end in main clauses; in subordinate clauses it stays attached to the verb, and in the infinitive with 'zu', 'zu' goes in between.
Der Maler trägt die zweite Farbschicht auf.
The painter applies the second coat of paint.
The verb consists of 'auf' (on/up) and 'tragen' (to carry). The basic idea of carrying something onto a surface or toward someone helps explain many meanings, such as applying cream or serving food.
Bitte tragen Sie die Schutzcreme auf die Haut auf.
Please apply the protective cream to the skin.
When a task is assigned to someone, the person receiving the task is always in the dative. The pattern is: jemandem (dative) etwas (accusative) auftragen.
Der Lehrer hat uns aufgetragen, ein Gedicht auswendig zu lernen.
The teacher assigned us the task of memorizing a poem.
When applying a substance, the surface is often given with the preposition 'auf' and the accusative. The pattern is: etwas (acc.) auf etwas (acc.) auftragen.
Tragen Sie den Kleber nur auf eine Seite des Papiers auf.
Apply the glue only to one side of the paper.
The expression 'dick auftragen' (to lay it on thick) means 'to greatly exaggerate.' It is always used in this fixed form.
Wenn er von seinen Reisen erzählt, trägt er immer ziemlich dick auf.
When he talks about his travels, he always lays it on pretty thick.
In the meaning 'to make someone look thicker,' 'auftragen' (to add bulk) is often used without an object. The subject is then the item of clothing.
Ich mag das Muster, aber die Jacke trägt leider auf.
I like the pattern, but unfortunately the jacket adds bulk.
The meaning 'to serve food' is rather formal and is mainly used in gastronomy or on festive occasions.
Zum Dessert wurde ein Schokoladenmousse aufgetragen.
A chocolate mousse was served for dessert.
to transmit / transfer
'übertragen' and 'auftragen' can both mean that a person receives a task or responsibility. 'übertragen' sounds more like passing on responsibility, rights, or areas of authority, often in an official or factual context. 'auftragen' is used when someone tells another person to do something. There is often a clear role involved: a boss, teacher, parents, or another authority figure. Learners may confuse the words because both have to do with tasks. A practical guide: you 'überträgt Verantwortung' (transfer responsibility), but you 'trägt jemandem eine konkrete Aufgabe auf' (assign someone a specific task).

Die Chefin trägt ihrem Assistenten auf, die Reise morgen zu buchen.
The boss tells her assistant to book the trip tomorrow.
Die Leitung überträgt Anna die Verantwortung für das neue Projekt.
The management transfers responsibility for the new project to Anna.