
ausziehen
to move out
To permanently leave an apartment, house, or office in order to live or work somewhere else.
Meine Nachbarn ziehen nächste Woche aus.
My neighbors are moving out next week.
to move out / take off
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to move out
To permanently leave an apartment, house, or office in order to live or work somewhere else.
Meine Nachbarn ziehen nächste Woche aus.
My neighbors are moving out next week.

to take off
To remove one or more items of clothing from the body.
Bitte zieh deine schmutzigen Schuhe aus, bevor du reinkommst.
Please take off your dirty shoes before you come in.

to undress
To remove all of one's clothing; to get naked.
Er zog sich aus und sprang in den See.
He undressed and jumped into the lake.

to extend
To lengthen or enlarge an object by pulling, for example a table or a telescope.
Kannst du bitte den Tisch ausziehen? Wir erwarten heute Abend mehr Gäste.
Can you please extend the table? We're expecting more guests this evening.
'Ausziehen' is a separable verb. The prefix 'aus-' comes at the end of the sentence in a main clause. In subordinate clauses or in the infinitive with 'zu', the verb stays together.
Er zieht seine Jacke aus.
He takes off his jacket.
The basic meaning is 'etwas herausziehen' (to pull something out). This logic connects all the meanings: you 'pull yourself' out of an apartment (move out), you pull clothing off the body (take off clothes), or you pull out part of a piece of furniture (make it larger).
Er zog den Schlüssel aus dem Schloss.
He pulled the key out of the lock.
The exact meaning becomes clear from the object or the grammar. An item of clothing as the object means 'to take off'. A piece of furniture as the object means 'to extend'. The reflexive pronoun 'sich' means 'to undress'. Without an object in the context of an apartment, it means 'to move out'.
Zieh bitte deine Schuhe aus!
Please take off your shoes!
When saying where someone is moving out from, you use the preposition 'aus' with the dative. This structure is very common.
Sie ist letzten Monat aus ihrer alten Wohnung ausgezogen.
She moved out of her old apartment last month.
There is a difference: 'sich ausziehen' usually means to take off all one's clothes. 'Etwas ausziehen' (with an accusative object) means to take off only a specific item of clothing.
Er zog sich komplett aus, um zu duschen.
He undressed completely in order to shower.
to put down / take off
Ablegen and ausziehen can both refer to removing an item of clothing. Ausziehen emphasizes the movement away from the body, while ablegen, especially with a coat, hat, or scarf, can also suggest putting it aside. Use ausziehen for the action of removing clothing from the body and ablegen when putting the item down is the focus.

Im Flur zog sie nach dem Spaziergang ihren nassen Mantel aus.
After her walk, she took off her wet coat in the hallway.
Im Flur legte sie nach dem Spaziergang ihren nassen Mantel ab.
After her walk, she took off her wet coat in the hallway.