ablegen
to put down
To place something that you are holding or carrying in a specific place.
Du kannst deine Tasche hier auf dem Stuhl ablegen.
You can put your bag down here on the chair.

to put down / take off
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to put down
To place something that you are holding or carrying in a specific place.
Du kannst deine Tasche hier auf dem Stuhl ablegen.
You can put your bag down here on the chair.

'Ablegen' is a separable verb. The prefix 'ab-' goes at the end in a main clause. In subordinate clauses and in the infinitive with 'zu', the verb stays together.
Legen Sie den Mantel bitte hier ab.
Please take off your coat here.
The prefix 'ab-' often carries the meaning of removal, separation, or completion. This idea appears in many meanings of 'ablegen' (to put down/take off), from putting down an object to giving up a habit.
Er hat seine alte Brille abgelegt und trägt jetzt Kontaktlinsen.
He has taken off his old glasses and now wears contact lenses.
In the meaning 'Kleidung ausziehen' (to take off clothing), 'ablegen' is more polite and formal than other verbs. You often hear it at a coat check in a theater or restaurant.
Darf ich Ihnen helfen, den Mantel abzulegen?
May I help you take off your coat?
The combination 'eine Prüfung ablegen' (to take an exam) is a fixed expression and means to take part in a test. You can also use 'ablegen' with ein Examen (an exam), einen Test (a test), or eine Klausur (a written exam).
Alle Studierenden müssen am Ende des Semesters eine Prüfung ablegen.
All students have to take an exam at the end of the semester.
When it is about stopping a way of behaving, 'ablegen' is used with abstract nouns such as 'Angst' (fear), 'Zweifel' (doubts), 'eine schlechte Angewohnheit' (a bad habit), or 'ein Akzent' (an accent).
Es ist schwer, eine schlechte Angewohnheit abzulegen.
It is hard to give up a bad habit.
In an office context, 'ablegen' almost always means to sort and save documents or files. Typical objects are 'Akten' (files), 'Dokumente' (documents), or 'E-Mails' (emails).
Könntest du bitte diese Dokumente für mich ablegen?
Could you please file these documents for me?
In formal or legal contexts, 'ablegen' is used in fixed expressions such as 'ein Geständnis ablegen' (to make a confession) or 'einen Eid ablegen' (to take an oath). It means to make an official statement.
Der Angeklagte weigerte sich, ein Geständnis abzulegen.
The defendant refused to make a confession.
In the meaning 'a ship starts/departs', 'ablegen' is used without an object. The subject is the ship or boat itself.
Die Fähre nach Schweden legt in einer Stunde ab.
The ferry to Sweden departs in one hour.
to pass / consist / exist
'bestehen' and 'ablegen' can both have to do with exams, but they describe different things. 'eine Prüfung ablegen' means that you take part in the exam. It does not yet say for sure whether the result is good. 'eine Prüfung bestehen' means that you successfully pass the exam. So you can take an exam and then pass it or fail it. Learners easily confuse the words because both often appear together with 'Prüfung' (exam). Memory tip: First you 'legst die Prüfung ab' (take the exam), then you find out whether you 'bestanden' (passed) it.

Wir haben gestern die mündliche Prüfung abgelegt.
We took the oral exam yesterday.
Sie hat die Fahrprüfung beim zweiten Versuch bestanden.
She passed the driving test on her second attempt.