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.

'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 report
'berichten' and 'erstatten' can both mean that you pass on information about a situation. 'berichten' is the normal and more general word: you can report or tell friends, family, colleagues, or the media about something. In this meaning, 'erstatten' fits almost only in fixed or formal expressions, especially 'Bericht erstatten' (to give a report). This sounds more official and often as if you have to give information to a superior or an authority. Learners may confuse the words because 'Bericht erstatten' is similar to 'berichten'. In everyday life, however, 'berichten' is usually more natural; 'Bericht erstatten' is used more in official situations.

Der Mitarbeiter musste der Chefin am Morgen Bericht über den Unfall erstatten.
In the morning, the employee had to report to the boss about the accident.
Ich habe meiner Schwester nach der Feier berichtet, wie das Gespräch gelaufen ist.
After the party, I told my sister how the conversation had gone.