mahnen
to remind
To firmly remind someone of a payment that is due, a debt, or another obligation, often in written form.
Die Bibliothek hat mich gemahnt, die überfälligen Bücher zurückzugeben.
The library reminded me to return the overdue books.

to remind / warn
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to remind
To firmly remind someone of a payment that is due, a debt, or another obligation, often in written form.
Die Bibliothek hat mich gemahnt, die überfälligen Bücher zurückzugeben.
The library reminded me to return the overdue books.

'Mahnen' (to remind / warn) is a formal word that is often used in written language, especially in official or business contexts.
Das Unternehmen mahnt seine Kunden, die Rechnungen pünktlich zu bezahlen.
The company reminds its customers to pay their invoices on time.
The verb 'mahnen' (to remind / warn) is closely related to the noun 'die Mahnung' (reminder notice). A Mahnung is the written document you receive when you are formally reminded to pay or fulfill an obligation.
Ich habe heute eine Mahnung bekommen, weil ich eine Rechnung vergessen habe.
I received a payment reminder today because I forgot an invoice.
In the meaning 'to urgently call on someone to do something', 'mahnen' (to warn / urge) is often used with the preposition 'zu' + dative. You urge someone toward a particular behavior.
Der Arzt mahnte den Patienten zu mehr Bewegung.
The doctor urged the patient to get more exercise.
When something recalls an event, the fixed structure 'mahnen an' (to serve as a reminder of / commemorate) + accusative is used. The subject here is often a memorial, a place, or a symbol.
Die Ruinen der alten Burg mahnen an eine zerstörerische Schlacht.
The ruins of the old castle serve as a reminder of a destructive battle.
to remember / remind
'erinnern' and 'mahnen' both have to do with reminding, but 'mahnen' is much stronger. With 'jemanden erinnern an' (to remind someone of/about), you usually say neutrally that someone should not forget something. 'mahnen' is used when a duty, a payment, or serious behavior is important. It sounds like pressure, a warning, or an official request. In addition, something can 'mahnen an' (serve as a reminder of / commemorate) a negative event; that sounds more serious than 'erinnern an'. Learners confuse the words because 'mahnen' is often also a kind of reminding. As a practical rule: for normal appointments and small tasks, 'erinnern' fits. For debts, rules, danger, or moral warnings, 'mahnen' is more appropriate.

Die Firma mahnt den Kunden, weil die Rechnung seit zwei Wochen offen ist.
The company sends the customer a reminder because the invoice has been unpaid for two weeks.
Der Mitarbeiter erinnert seine Kollegin an den Termin vor dem Wochenende.
The employee reminds his colleague about the appointment before the weekend.