Payoff amount is how much you would actually have to pay to satisfy the debt. It is not the same thing as the current balance that shows up on your statement, but is not necessarily the amount you owe. This is not obvious, but the difference between current balance and payoff amount is crucial when you are ready to pay off a debt.
Current balance means the amount you owe as of the date of the statement. As of the day after the statement, you owe more. In other words, if you are trying to pay off a credit card, and the statement says your balance is $514, you may not be able to bring your balance to zero by writing a check for $514. Instead, you would need to contact the lender to find out your payoff amount.
The payoff amount is really just a more-current balance number. But if you are trying to eliminate a debt, you need to pay it all off. If you just pay the current balance, you may be left with a few cents or dollars left in the account. Over time, that could become more than an irritation; it could become a significant obstacle to eliminating your debt.
So before you pay off a debt, call the lender to find out exactly how much you owe, and how much it will take to pay the debt off, in full.