sich etwas wünschen
to wish for
To have a wish or desire for something that one would like to receive or achieve.
Zu meinem Geburtstag wünsche ich mir ein neues Buch.
For my birthday, I wish for a new book.

to wish
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to wish for
To have a wish or desire for something that one would like to receive or achieve.
Zu meinem Geburtstag wünsche ich mir ein neues Buch.
For my birthday, I wish for a new book.

To express a wish that does not correspond to reality, 'wünschen' (to wish) is often used in the subjunctive II ('ich wünschte' (I wish)). The subordinate clause that describes the wish is then also in the subjunctive II.
Ich wünschte, ich könnte besser Klavier spielen.
I wish I could play the piano better.
When you wish for something for yourself, the verb is reflexive. The structure is: subject + sich (dative) + desired thing (accusative).
Sie wünscht sich ein Fahrrad zum Geburtstag.
She wishes for a bicycle for her birthday.
When you wish another person something good, you need a dative object (the person) and an accusative object (the good thing).
Wir wünschen unseren Nachbarn einen schönen Urlaub.
We wish our neighbors a nice vacation.
In the service sector, the question 'Was wünschen Sie?' (What would you like?) is used to ask about a customer's wishes in a particularly polite and formal way.
Guten Tag, was wünschen Sie? – Ich hätte gern einen Kaffee.
Good day, what would you like? – I'd like a coffee.
to hope
'hoffen' and 'wünschen' can both express that you want something positive. 'hoffen' is used when the outcome is uncertain: you want something to happen. 'sich etwas wünschen' places more emphasis on your own wish or desire to receive or achieve something. 'jemandem etwas wünschen' means saying something good to another person, for example wishing them luck or success. Learners may confuse the words because both have to do with wishes. Memory tip: you wish for something, but you hope that it really happens.

Zum Geburtstag wünscht sich Lena ein Fahrrad.
For her birthday, Lena wishes for a bicycle.
Heute hofft er, dass die neue Wohnung ruhig ist.
Today he hopes that the new apartment is quiet.