
erstatten
to reimburse
To give someone back money that he or she spent for a specific purpose.
Die Firma erstattet allen Mitarbeitern die Reisekosten für Dienstreisen.
The company reimburses all employees for travel expenses on business trips.
to reimburse / report
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to reimburse
To give someone back money that he or she spent for a specific purpose.
Die Firma erstattet allen Mitarbeitern die Reisekosten für Dienstreisen.
The company reimburses all employees for travel expenses on business trips.

to report a crime
To officially report a criminal offense to the police.
Nach dem Einbruch hat sie sofort bei der Polizei Anzeige erstattet.
After the break-in, she immediately reported it to the police.

to report
To give someone, usually a superior or institution, a formal oral or written account of a matter.
Der Spion musste seiner Führung regelmäßig Bericht erstatten.
The spy had to report regularly to his handler.
'Erstatten' is used almost only in fixed expressions. The most important ones are 'Anzeige erstatten' (to report a crime, to the police), 'Bericht erstatten' (to report to a superior), and 'Kosten/Geld erstatten' (to reimburse costs/money).
Nach dem Diebstahl musste er bei der Polizei Anzeige erstatten.
After the theft, he had to report the crime to the police.
The verb 'erstatten' belongs to formal administrative and business language. You often find it in official documents, at the police, or in communication with companies.
Bitte reichen Sie die Belege ein, damit wir Ihnen die Kosten erstatten können.
Please submit the receipts so that we can reimburse you for the costs.
When costs are reimbursed, the person who gets the money back is in the dative. Die Firma (subject) erstattet dem Mitarbeiter (dative) die Kosten (accusative).
Mein Arbeitgeber erstattet mir die Fahrtkosten für die Dienstreise.
My employer reimburses me for the travel costs for the business trip.
The expression 'Anzeige erstatten' (to report a crime) is often used with the preposition 'gegen' (against) for the accused person and 'bei' (at/with) for the institution, usually the police.
Sie hat bei der Polizei Anzeige gegen Unbekannt erstattet.
She filed a complaint with the police against an unknown person.
to replace / reimburse
Ersetzen generally refers to financially compensating for damage, a loss, or an expense. Erstatten is especially appropriate when someone gets back money for specific costs they have already paid. For such expenses, erstatten is usually the more common choice.

Die Fluggesellschaft erstattet mir die bereits bezahlten Kosten für das Ticket.
The airline is reimbursing me for the ticket costs I have already paid.
Die Versicherung ersetzt mir den finanziellen Verlust durch den Flugausfall.
The insurance company is compensating me for the financial loss caused by the canceled flight.