renunciation / waiver
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renunciation
The voluntary or forced decision to give something up, such as a claim, a right, or a pleasure.
Der Verzicht auf sein Erbe war eine schwere Entscheidung für ihn.
The renunciation of his inheritance was a difficult decision for him.
'Der Verzicht' (renunciation/waiver) is the noun that belongs to the verb 'verzichten' (to do without / to give up). When you do without something, the result is 'der Verzicht' (renunciation).
Ich verzichte heute auf den Nachtisch.
I'm skipping dessert today.
The noun is almost always used with the preposition 'auf' and the accusative: 'der Verzicht auf etwas' (the renunciation of something / doing without something).
Der Verzicht auf Plastiktüten ist ein wichtiger Schritt für die Umwelt.
Doing without plastic bags is an important step for the environment.
A common construction describes how hard or easy the action is for a person: 'Der Verzicht fällt jemandem (dative) schwer/leicht' (giving something up is hard/easy for someone).
Nach so vielen Jahren fiel ihm der Verzicht auf das Rauchen sehr schwer.
After so many years, giving up smoking was very hard for him.
victim / sacrifice
'das Opfer' and 'der Verzicht' can both mean that you do not take something important or that you give it up. The difference lies in the point of view. 'der Verzicht' generally describes giving something up, for example money, free time, or a right. 'ein Opfer bringen' (to make a sacrifice) sounds stronger: you give up something valuable, usually for a goal or for another person. Learners may confuse the words because in both cases something is lost. 'Verzicht' is more neutral, while 'Opfer' emphasizes more strongly that it is difficult or personally important.
Der Verzicht auf Süßigkeiten fällt ihm diese Woche besonders schwer.
Giving up sweets is especially hard for him this week.
Für das Studium brachte sie viele Opfer und arbeitete auch am Wochenende.
She made many sacrifices for her studies and also worked on weekends.