
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.

to take off
To take off an item of clothing, especially a coat, hat, or scarf.
Bitte legen Sie Ihren Mantel an der Garderobe ab.
Please take off your coat at the coat check.

to take an exam
To take part in an exam or test and complete it, usually successfully.
Nächste Woche muss ich meine Fahrprüfung ablegen.
Next week I have to take my driving test.

to lose
To give up and stop a habit, feeling, or way of behaving.
Er konnte seine Schüchternheit nie ganz ablegen.
He was never able to completely lose his shyness.

to file
To systematically organize documents, files, or data and keep them in a designated place.
Die Rechnungen werden digital in einem Ordner abgelegt.
The invoices are filed digitally in a folder.

to make a confession/oath
To make a formal, often legally binding statement, for example in court or when being sworn in.
Der Zeuge musste vor Gericht einen Eid ablegen.
The witness had to take an oath in court.

to depart
In shipping: to leave the anchorage or dock and begin the journey.
Das Schiff wird pünktlich um 18 Uhr ablegen.
The ship will depart on time at 6 p.m.
'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 do / make
With „machen“, you can say in general that you are taking an exam; the phrasing is neutral and very common in everyday language. „Ablegen“ refers to the same process but sounds more fixed and somewhat formal. That is why official information usually uses „eine Prüfung ablegen“, while conversation often uses „eine Prüfung machen“.

Morgen lege ich die Prüfung für meinen Sprachkurs ab.
I’m taking the exam for my language course tomorrow.
Morgen mache ich die Prüfung für meinen Sprachkurs.
I’m taking the exam for my language course tomorrow.