A record of what you've sold to a customer. You send an invoice to your customers so they know what they need to pay you. Let's say you own a pirate supply store. You would send Captain Hannah the Horrible an invoice for the twelve pirate flags, 100 cannonballs, and three parrots she bought last week.
Note: Don't confuse invoices with bills. In QuickBooks, bills are what your vendors send you for you to pay them.