aufhalten
to delay
To prevent someone or something from continuing on or continuing an activity; to delay.
Entschuldigung für die Verspätung, ich wurde von einem Anruf aufgehalten.
Sorry for being late; I was delayed by a phone call.

to delay / stay / hold open
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to delay
To prevent someone or something from continuing on or continuing an activity; to delay.
Entschuldigung für die Verspätung, ich wurde von einem Anruf aufgehalten.
Sorry for being late; I was delayed by a phone call.

'Aufhalten' (to delay/to hold open) is a separable verb. The prefix 'auf-' goes to the end in a main clause. In subordinate clauses and in the infinitive with 'zu', the verb stays together.
Der Stau hält uns auf.
The traffic jam is delaying us.
The reflexive form 'sich aufhalten' (to stay/be located) always means that someone is in a place. Without the reflexive pronoun 'sich', the verb means 'to delay' or 'to hold open'.
Wo hält sich der Verdächtige gerade auf?
Where is the suspect currently staying?
In the meaning 'to delay', the structure 'jemanden aufhalten' (to delay someone) is often used. In the passive, the form is 'aufgehalten werden' (to be delayed).
Lass uns gehen, ich will dich nicht länger aufhalten.
Let's go; I don't want to delay you any longer.
The form 'sich aufhalten' (to stay/be located) is always used with an indication of place. You stay 'in' a city, 'an' a place, or 'bei' friends.
Im Sommer halten wir uns am liebsten am See auf.
In summer, we prefer to stay by the lake.
'Sich aufhalten' (to stay/be located) is often part of formal or official language, for example in police reports or administrative letters.
Der Gesuchte soll sich im Ausland aufhalten.
The wanted person is said to be staying abroad.
In the meaning 'to hold open', the objects are almost always 'die Tür' (the door), 'das Fenster' (the window), or 'das Tor' (the gate).
Sei so nett und halte mir die Tür auf, meine Hände sind voll.
Be so kind and hold the door open for me; my hands are full.
to stay / remain
'bleiben' and 'sich aufhalten' can both mean that someone is in a place and does not leave right away. 'bleiben' sounds everyday and often emphasizes the decision not to leave. 'sich aufhalten' sounds a bit more precise or formal and more often describes where someone is for a certain period of time. In everyday life, people often say 'Ich bleibe hier' (I'm staying here). In official or more precise situations, 'sich aufhalten' works well, for example when giving a location.

Der Besucher hält sich seit dem Morgen im Büro auf.
The visitor has been staying in the office since this morning.
Wegen des starken Regens bleiben wir heute zu Hause.
Because of the heavy rain, we're staying at home today.