aufgeben
to give up
To stop trying or continuing something because one does not expect success.
Obwohl das Rennen sehr anstrengend war, hat sie nicht aufgegeben.
Although the race was very exhausting, she did not give up.

to give up / submit
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to give up
To stop trying or continuing something because one does not expect success.
Obwohl das Rennen sehr anstrengend war, hat sie nicht aufgegeben.
Although the race was very exhausting, she did not give up.

'Aufgeben' (to give up / submit) is a separable verb. The prefix 'auf' comes at the end in a main clause. In subordinate clauses and in the infinitive with 'zu', the verb stays together.
Ich gebe diesen Plan nicht auf.
I am not giving up this plan.
The verb is made up of the prefix 'auf' and the verb 'geben' (to give). The basic idea of giving something away from oneself or letting go of something can be recognized in many meanings.
Er musste seinen Traum vom Profisport aufgeben.
He had to give up his dream of professional sports.
In the meaning 'to surrender' or 'to stop continuing', 'aufgeben' (to give up) can also be used without an accusative object. This does not apply to the other meanings.
Kurz vor dem Ziel war er so erschöpft, dass er aufgab.
Shortly before the finish line, he was so exhausted that he gave up.
In the context of 'handing over', 'aufgeben' (to send / submit) is often combined with things that one hands in or checks in at an institution. Typical phrases are: ein Paket aufgeben (to send a package), einen Brief aufgeben (to mail a letter), Gepäck aufgeben (to check in luggage), or eine Bestellung aufgeben (to place an order).
Am Flughafen müssen Sie Ihr Gepäck am Schalter aufgeben.
At the airport, you have to check in your luggage at the counter.
When a teacher assigns tasks, the structure 'jemandem (Dativ) etwas (Akkusativ) aufgeben' (to assign someone something) is used. This meaning is common almost only in school or university contexts.
Der Professor hat uns eine schwere Aufgabe aufgegeben.
The professor assigned us a difficult task.
to put / to set
'stellen' and 'aufgeben' can both mean that you pass something on officially or for processing. However, they are used with different nouns. You 'stellt' an application, especially with an authority or institution: 'einen Antrag stellen' (to submit an application). You 'gibt ... auf' an order, a letter, or a package when it is supposed to be processed or sent. Learners may confuse the words because in both cases something goes to some office or place. The fixed expression is important: 'einen Antrag stellen', but 'eine Bestellung aufgeben' (to place an order) or 'ein Paket aufgeben' (to send a package).

Nach dem Gespräch gibt er die Bestellung direkt im System auf.
After the conversation, he enters the order directly into the system.
Für die neue Wohnung muss Maria noch einen Antrag stellen.
Maria still has to submit an application for the new apartment.