aufhängen
to hang up
To attach something to a hook, line, or nail so that it hangs down freely.
Kannst du mir helfen, die Wäsche aufzuhängen?
Can you help me hang up the laundry?

to hang up
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to hang up
To attach something to a hook, line, or nail so that it hangs down freely.
Kannst du mir helfen, die Wäsche aufzuhängen?
Can you help me hang up the laundry?

'Aufhängen' (to hang up) is a separable verb. The prefix 'auf-' comes at the end in a main clause, stays attached to the verb in a subordinate clause, and in the infinitive with 'zu', 'zu' goes in between.
Er hängt das neue Bild im Wohnzimmer auf.
He hangs the new picture up in the living room.
The meaning depends strongly on the grammar. If there is an accusative object, it usually means to attach something. If the verb is reflexive ('sich aufhängen'), it refers either to a technical problem or to suicide.
Ich hänge meine Jacke auf.
I hang up my jacket.
In this meaning, 'aufhängen' (to hang up) is often combined with things that you attach around the home: 'Wäsche aufhängen' (to hang up laundry), 'ein Bild aufhängen' (to hang up a picture), 'einen Mantel aufhängen' (to hang up a coat), or 'Gardinen aufhängen' (to hang up curtains).
Nach dem Waschen hänge ich die Wäsche immer im Garten auf.
After washing, I always hang the laundry up in the garden.
When it is about a phone call, 'aufhängen' (to hang up) is often used without an object. It then stands at the end of the sentence and signals the end of the conversation.
Er hat einfach aufgelegt, ohne sich zu verabschieden.
He simply hung up without saying goodbye.
The reflexive form 'sich aufhängen' in this meaning is used only for technical devices or software. The subject is always something non-human, such as 'der Computer' (the computer), 'das Programm' (the program), or 'der Browser' (the browser).
Ich muss den Laptop neu starten, weil das Programm sich aufgehängt hat.
I have to restart the laptop because the program has frozen.
to hang
'hängen' and 'aufhängen' are very close in meaning when you put something on a hook, nail, or line. 'aufhängen' clearly describes the action: someone fastens something so that it hangs afterward. 'hängen' can also mean this action, but very often it describes the state: 'Das Bild hängt schon an der Wand' (the picture is already hanging on the wall). That is why the words can easily be confused. As a practical tip: for the action, 'aufhängen' is often more precise; for the state, 'hängen' is more natural.

Zum Trocknen hängt er die Handtücher auf dem Balkon auf.
He hangs the towels up on the balcony to dry.
Die Ärztin hängt ihren Mantel an die Garderobe.
The doctor hangs her coat on the coat rack.