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.

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' both have to do with something in the future. 'warten' means letting time pass until someone comes or something happens. It is often used with 'auf': 'auf den Bus warten' (to wait for the bus), 'auf eine Antwort warten' (to wait for an answer). 'erwarten' more often means that you expect something to happen, or that you demand or hope for something from someone. You can also 'Gäste erwarten' (expect guests); this sounds somewhat more formal than 'auf Gäste warten' (to wait for guests). The risk of confusion lies in the fact that both words are connected with hope or the future. Memory tip: 'warten' describes the situation; 'erwarten' describes the assumption or demand.

Vor dem Termin wartet mein Nachbar auf den Arzt.
Before the appointment, my neighbor waits for the doctor.
Für den Kurs erwartet die Lehrerin pünktliche Teilnahme.
For the course, the teacher expects participants to be on time.