When you receive a notice in the mail that you owe money, the usual name for that notice is "a bill."
Every month we get bills for our electricity, our telephone and Internet service, and our gas service. (We heat our house with gas that is piped to our home).
We also receive insurance bills and tax bills.
We are billed for many services.
We need to pay our bills. We have a good credit rating because we pay our bills on time.
If I get a bill from a doctor that I don't understand, I call the doctor's number; a recorded voice says "Press 1 to renew a prescription, 2 to make an appointment, 3 for billing."
We got a bill with a mistake in it. There was a charge on the bill for a service that hadn't been performed. We called the company to dispute the bill. The company said, "Oh, we're sorry, we'll take it off your bill. Don't pay the bill we sent you, we'll send you a new one."
When we're finished eating a meal in a restaurant, it is more common to tell the waiter "please bring us the check" but sometimes we say "please bring us the bill;" it means the same thing.
Unfortunately, the word "bill" has lots of other meanings, but I won't try to explain them.