
der Wille
will
The mental strength or firm intention to make a decision and pursue a goal.
Wo ein Wille ist, ist auch ein Weg.
Where there’s a will, there’s a way.
will
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will
The mental strength or firm intention to make a decision and pursue a goal.
Wo ein Wille ist, ist auch ein Weg.
Where there’s a will, there’s a way.

last will
A legal document (will) in which a person determines what should happen to their assets and property after their death.
In seinem letzten Willen hat er das Haus seiner Tochter vermacht.
In his last will, he left the house to his daughter.

for the sake of
A fixed expression that expresses a reason or purpose, often in the sense of 'for the sake of' or 'because of'.
Um des Friedens willen sollten wir einen Kompromiss finden.
For the sake of peace, we should find a compromise.
'Der Wille' (will) is the noun related to the verb 'wollen' (to want). It describes the inner strength or intention behind the desire expressed by 'wollen'.
Er will das Projekt beenden, aber ihm fehlt der Wille, wirklich anzufangen.
He wants to finish the project, but he lacks the will to really get started.
Common collocations are 'einen starken Willen haben' (to have a strong will), 'seinen Willen durchsetzen' (to get one's way), or 'gegen seinen Willen' (against one's will). 'Die Kraft des Willens' (the power of will) is also a fixed combination.
Sie hat einen eisernen Willen und erreicht immer ihre Ziele.
She has an iron will and always achieves her goals.
The combination 'letzter Wille' (last will) is a fixed legal term for a will. The adjective 'letzter' (last) is the key clue to the meaning here.
Der Notar verlas den letzten Willen des Verstorbenen.
The notary read out the last will of the deceased.
The expression 'um … willen' surrounds a noun in the genitive. It expresses a reason or purpose and is typical of somewhat elevated language.
Um der Gerechtigkeit willen muss der Fall neu untersucht werden.
For the sake of justice, the case must be investigated again.
wish
Both words can express that someone wants something specific. A „Wunsch“ tends to describe hope for an outcome, while „Wille“ refers to a firm intention that guides one’s actions. Use „Wunsch“ when something is merely hoped for, and „Wille“ when someone intends to pursue it resolutely.

Sein Wille, den Wettbewerb zu gewinnen, war deutlich zu erkennen.
His determination to win the competition was clearly evident.
Ihr Wunsch ist, dass alle Gäste pünktlich ankommen.
Her wish is for all the guests to arrive on time.