die Rechnung
bill / invoice
A document that lists the costs for goods received or services provided and requests payment.
Herr Ober, könnten Sie uns bitte die Rechnung bringen?
Waiter, could you please bring us the bill?

bill / calculation
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bill / invoice
A document that lists the costs for goods received or services provided and requests payment.
Herr Ober, könnten Sie uns bitte die Rechnung bringen?
Waiter, could you please bring us the bill?

'Die Rechnung' (the bill/calculation) is derived directly from the verb 'rechnen' (to calculate). This connection helps you understand the basic meanings 'payment request' (m01) and 'calculation' (m02).
Ich rechne kurz aus, was jeder bezahlen muss.
I'll quickly calculate how much everyone has to pay.
You can recognize the meaning from the context: if it's about money, paying, or services, the payment request (m01) is meant. If it's about logic, planning, or numbers, the calculation (m02) is meant.
Die Rechnung für die Autoreparatur war sehr hoch.
The bill for the car repair was very high.
In the context of a payment request, typical verbs are 'eine Rechnung bezahlen' (to pay a bill), 'eine Rechnung begleichen' (to settle a bill, more formal), 'eine Rechnung ausstellen' (to issue an invoice), or 'eine Rechnung bekommen' (to receive a bill).
Bitte begleichen Sie die Rechnung innerhalb von 14 Tagen.
Please settle the invoice within 14 days.
In the sense of a calculation, there is the fixed expression 'die Rechnung geht auf' (the calculation works out). It means that a plan works or an outcome occurs as expected.
Wir haben weniger ausgegeben als geplant. Unsere Rechnung ist voll aufgegangen.
We spent less than planned. Our calculation worked out perfectly.
The fixed expression 'Rechnung tragen' (to take into account) requires the dative. You take 'etwas' (something) or 'jemandem' (someone) into account.
Bei der Planung müssen wir den neuen Umständen Rechnung tragen.
In our planning, we must take the new circumstances into account.
The expression 'einen Strich durch die Rechnung machen' (to thwart someone's plans) also uses the dative. You thwart 'jemandem' (someone's) plans.
Ich wollte am Samstag ausschlafen, aber der laute Nachbar hat mir einen Strich durch die Rechnung gemacht.
I wanted to sleep in on Saturday, but the loud neighbor thwarted my plans.
account
'die Rechnung' and 'das Konto' are often used in connection with money, but they mean very different things. A 'Rechnung' is a document with costs that you are supposed to pay. A 'Konto' is a place where money is managed, for example at a bank. In addition, 'Konto' can also mean an account for an online service. Learners sometimes confuse the words because you often pay a bill from an account. In practical terms: you receive 'die Rechnung'; you use 'das Konto' to pay or log in.

Zu Hause haben die Eltern gemerkt, dass sie die Rechnung für das neue Fahrrad noch nicht bezahlt haben.
At home, the parents noticed that they had not yet paid the bill for the new bicycle.
Nach der Arbeit hat Max sein Konto geprüft, weil die Miete abgebucht wurde.
After work, Max checked his account because the rent had been debited.