to return / to give back
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to return
To give something back to someone that you borrowed, received, or took from them. This also includes returning purchased goods to a store.
Ich muss dir dein Buch morgen unbedingt zurückgeben.
I absolutely have to return your book to you tomorrow.
'Zurückgeben' (to return / to give back) is a separable verb. The prefix 'zurück-' comes at the end in a main clause, while it stays attached to the verb in a subordinate clause.
Er gibt mir das Geld morgen zurück.
He will give me the money back tomorrow.
The verb is built very logically from the adverb 'zurück' (back) and the verb 'geben' (to give). The literal meaning is 'to give something in the opposite direction.'
Kannst du mir bitte meinen Stift zurückgeben?
Could you please give me my pen back?
The verb is typically used with a person in the dative (to whom one gives something) and a thing in the accusative (what one gives).
Ich gebe dir das Buch nächste Woche zurück.
I'll give you the book back next week.
In this meaning, 'zurückgeben' (to give back/throw up) is a euphemism, meaning a softened and indirect paraphrase for 'sich übergeben' (to vomit). It is used more colloquially.
Dem Kind wurde auf der langen Autofahrt schlecht und es musste alles zurückgeben.
The child felt sick on the long car ride and had to throw everything up.
to keep
'behalten' and 'zurückgeben' are opposites when you have received or borrowed something from someone. 'behalten' means that you do not give it back and it stays with you. 'zurückgeben' means that you give it back to the person or to the store. The contrast is especially clear with borrowed things or purchased goods: you decide whether to keep them or return them.
Meine Mutter gibt das kaputte Ladegerät morgen im Laden zurück.
My mother is returning the broken charger to the store tomorrow.
Den Pullover behalte ich, weil er mir gut passt.
I'm keeping the sweater because it fits me well.