Bill and invoice both refer to a document that is requiring money to be paid for goods or services provided.
In everyday conversation, we usually talk about bills:
The electrical company sends you an electrical bill.
The phone company sends you a phone bill.
After you get treated in a hospital, if you don't have health insurance, you'll have a lot of medical bills to pay.
When talking about a business or professional relationship, then we usually use the word invoice:
A consultant who gave advice to a company would then send the company an invoice (requiring payment for his time).
When a factory sends products to a store, it will also send the store an invoice requiring payment for those products.
Both bills and invoices are sent BEFORE payment. They are requiring the payment to be made. A receipt, on the other hand, is sent AFTER payment. It is the confirmation and proof that the payment was made.
After you buy something in a supermarket, you get a receipt.
After the store pays the factory for the products, the factory will provide a receipt.