Bill is one of those words that have many meanings.
Yes, in the US, it is the standard description for a banknote:
a five-dollar bill
This is really specific to American English; in the U.K., you'd say "a ten pound note." (My answer is really based on American English.)
In crime movies, when someone demands a ransom, you can often hear:
a million dollars in small bills
Here, the Lifehacker website advises people to always carry $20 with them in small bills.
And yes, bills are those "invoices" you get every month for your utilities like electricity and the like. You could hear a phrase like:
He had to get a better-paying job because he had trouble even paying his bills.
When you hear "pay your bills", it's about those "invoices." A phrase with the word "pay" where "bill" would mean "banknote" would have to be "pay with a bill." E. g.,
After we finished our meal, we paid with a hundred dollar bill. (again, here it is the "banknote")
At the restaurant, when you are done with your meal, you'd say to your friends,
Let's get the bill.
This is again an "invoice" from the restaurant to you stating the amount of money that you should pay. In the UK, one would say, "let's get the check."
Other meanings of the word bill are:
— a poster with an advertisement that you put on the walls (in New York City, you will often see on fences that, e.g., surround a construction site, signs saying "Post No Bills"—meaning "do not put up advertisements");
— a specific law or a draft law.