abrechnen
to bill
To calculate the costs for a service or product and arrange for or receive payment.
Der Handwerker wird die erledigten Arbeiten nächste Woche abrechnen.
The tradesman will bill for the completed work next week.

to bill / settle accounts
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to bill
To calculate the costs for a service or product and arrange for or receive payment.
Der Handwerker wird die erledigten Arbeiten nächste Woche abrechnen.
The tradesman will bill for the completed work next week.

'Abrechnen' (to bill / settle accounts) is a separable verb. In a main clause, the prefix 'ab' comes at the end. In subordinate clauses and in the infinitive with 'zu', the verb stays together.
Die Firma rechnet die Reisekosten am Monatsende ab.
The company settles the travel expenses at the end of the month.
The verb 'abrechnen' (to bill / settle accounts) is closely connected with the nouns 'die Abrechnung' (statement/settlement) and 'die Rechnung' (bill/invoice). When you 'abrechnen' something, you create a statement or settle a bill.
Nach dem Projekt müssen wir eine detaillierte Abrechnung erstellen.
After the project, we have to prepare a detailed statement of costs.
In the financial meaning, people often specify what is being billed, for example 'Kosten' (costs), 'Leistungen' (services), or 'Stunden' (hours), and with whom one settles the bill, for example 'mit dem Kunden' (with the customer) or 'mit der Versicherung' (with the insurance company).
Der Arzt rechnet die Behandlung direkt mit der Krankenkasse ab.
The doctor bills the treatment directly to the health insurance provider.
The expression 'mit jemandem abrechnen' (to settle scores with someone) has a figurative, negative meaning: to take revenge. The preposition 'mit' and a person in the dative are fixed parts of the expression here.
Er schwor, dass er eines Tages mit seinen Feinden abrechnen würde.
He swore that one day he would settle scores with his enemies.
When an amount is deducted from a sum, this is often expressed with the preposition 'von' (from). The sentence is often in the passive.
Die Vorauszahlung wird von der Endsumme abgerechnet.
The advance payment is deducted from the final total.
to pay
'bezahlen' and 'abrechnen' both occur in contexts involving costs, but they do not mean the same thing. 'bezahlen' means that money is given. 'abrechnen' means that costs are calculated, checked, or officially settled, often in a company, medical practice, or insurance context. Payment may follow, but the focus is on calculation and administration. Learners may confuse the words because money is often paid in the end. In everyday language, you say: 'Ich bezahle die Rechnung' (I pay the bill). A company or medical practice 'rechnet eine Leistung ab' (bills for a service).

Die Ärztin rechnet die Behandlung direkt mit der Krankenkasse ab.
The doctor bills the treatment directly to the health insurance company.
In der Pause bezahlt du das Parkticket am Automaten.
During the break, you pay for the parking ticket at the machine.