
warten auf
to wait for
To let time pass until someone arrives or something happens that you are expecting.
Ich warte schon seit zwanzig Minuten auf den Bus.
I've already been waiting for the bus for twenty minutes.
to wait / maintain
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to wait for
To let time pass until someone arrives or something happens that you are expecting.
Ich warte schon seit zwanzig Minuten auf den Bus.
I've already been waiting for the bus for twenty minutes.

to maintain
To regularly check and service technical devices, machines, or systems in order to keep them functioning.
Ein Techniker kommt einmal im Jahr, um die Heizung zu warten.
A technician comes once a year to maintain the heating system.

to be a long time coming
To take a long time before something or someone arrives or happens.
Eine Antwort auf seine Bewerbung ließ lange auf sich warten.
A response to his application was a long time coming.
The meaning of 'warten' depends strongly on the sentence structure. 'Warten auf' + accusative means to let time pass, to wait for something. 'Warten' + direct accusative (without 'auf') means to maintain a technical device.
Ich warte auf eine wichtige E-Mail.
I'm waiting for an important email.
When you are waiting for something or someone to come, you must use the preposition 'auf' followed by the accusative. The question is: 'Auf wen oder was wartest du?' (Who or what are you waiting for?).
Wir warten auf den Beginn des Films.
We're waiting for the start of the film.
In the technical meaning, 'warten' (to maintain) is a transitive verb that requires an accusative object. This object is always the machine or device being serviced.
Man sollte seine Klimaanlage regelmäßig warten lassen.
You should have your air conditioner maintained regularly.
The expression 'auf sich warten lassen' (to be a long time coming) means that something or someone takes a long time to happen or arrive. The subject of the sentence is the thing that is delayed.
Der Frühling hat dieses Jahr lange auf sich warten lassen.
Spring was a long time coming this year.
to expect
Warten and erwarten can both refer to a person’s arrival. Warten auf describes the time passing until the arrival, while erwarten focuses on the expected arrival. Use warten auf for the process and erwarten for the person who is expected.

Ich warte am Bahnhof auf meine Schwester, weil ihr Zug verspätet ist.
I am waiting for my sister at the station because her train is delayed.
Ich erwarte meine Schwester am Bahnhof, weil ihr Zug bald ankommt.
I expect my sister at the station because her train will arrive soon.