abhängen von
to depend on
To be determined or influenced by something or someone; to rely on something.
Ob wir den Ausflug machen, hängt ganz vom Wetter ab.
Whether we go on the trip depends entirely on the weather.

to depend / to hang out / to take down
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to depend on
To be determined or influenced by something or someone; to rely on something.
Ob wir den Ausflug machen, hängt ganz vom Wetter ab.
Whether we go on the trip depends entirely on the weather.

'Abhängen' is a separable verb. In a main clause, the prefix 'ab' stands at the end of the sentence or clause.
Ich hänge nach der Schule oft mit meinen Freunden ab.
I often hang out with my friends after school.
In the meaning of 'angewiesen sein' (to depend on / rely on), 'abhängen' is always used with the preposition 'von' and the dative. A very common structure is 'Es hängt davon ab, ob/wie/wann ...' (It depends on whether/how/when ...).
Der Erfolg des Projekts hängt von der guten Zusammenarbeit im Team ab.
The success of the project depends on good teamwork.
The meaning 'sich entspannen' (to relax) is colloquial and is used especially by younger people in informal situations. It often appears with the preposition 'mit' (with).
Was machst du so? Ich hänge nur ein bisschen auf dem Sofa ab.
What are you up to? I’m just hanging out on the sofa a bit.
When 'abhängen' means 'hinter sich lassen' (to leave behind), the verb needs an accusative object. You always leave behind 'jemanden' (someone) or 'etwas' (something).
Der Radfahrer konnte die Verfolger am letzten Berg abhängen.
The cyclist was able to leave the pursuers behind on the final mountain.
This meaning is the literal, physical action. It is used for things that you remove from a hook, a line, or a wall.
Kannst du mir helfen, die Gardinen abzuhängen?
Can you help me take down the curtains?
to hang
'hängen' and 'abhängen' can be opposites when it comes to things on the wall, on a line, or on a hook. With 'hängen', you put something up so that it hangs freely downward. With 'abhängen', you take down something that is already hanging. Learners may confuse the words because both are formed with 'hängen'. The important thing is the direction: 'hängen' puts something there; 'abhängen' takes it away.

Vor dem Streichen hängt er alle Bilder im Wohnzimmer ab.
Before painting, he takes down all the pictures in the living room.
Alina hängt das neue Poster über den Schreibtisch.
Alina hangs the new poster above the desk.