Balloon Payment

A balloon payment is a final loan payment that is larger than the regular periodic payments.

It arises in two ways. When a loan is repaid by level payments "for as long as necessary", the term is usually not a whole number of periods, and the small remainder can be paid together with the last regular payment, making that last payment larger than the others. A loan can also be scheduled with regular payments too small to repay it by the last date, so the final payment clears whatever is left.
Either way, the balloon payment is the Outstanding Balance just after the previous payment, accumulated for one period — the regular payment due then plus whatever the regular payments leave unpaid.
Contrast with a Drop Payment: the same remainder can instead be paid one period after the last regular payment, as a separate payment that is smaller than the regular one.
For a loan of $L$ with $n-1$ regular payments of $K$ followed by one balloon payment $B$ at time $n$ , at interest rate $i$ :

$B = L(1+i)^n - K \cdot s_{\overline{n-1}|i} \cdot (1+i)$

Or equivalently, $B$ is the Outstanding Balance $OB_{n-1}$ accumulated one more period: $B = OB_{n-1}(1+i)$ .

Balloon Payment on a Loan A borrower takes a $ $10{,}000$ loan at $8\%$ annual interest. The borrower makes payments of $ $1{,}200$ at the end of each year for 9 years, then clears the loan with a balloon payment at the end of year 10. Find the balloon payment.
Answer First find the outstanding balance after 9 payments using the retrospective method, with $(1.08)^9 = 1.999005$ and $s_{\overline{9}|8\%} = 12.487558$ :
 $\begin{align*} OB_9 &= 10000(1.08)^9 - 1200\,s_{\overline{9}|8\%} \\ &= 19990.05 - 14985.07 \\ &= \$5{,}004.98 \end{align*}$ 
The balloon payment at time 10 is:
 $\begin{align*} B &= OB_9(1.08) \\ &= 5004.98(1.08) \\ &= \$5{,}405.38 \end{align*}$ 
The final payment is more than four times the regular $ $1{,}200$ payment.

Balloon or Drop: Paying Off the Remainder A $ $10{,}000$ loan at $8\%$ annual interest is repaid by payments of $ $1{,}500$ at the end of each year for as long as necessary. Find how many full payments of $ $1{,}500$ can be made, and the final payment if the remainder is added to the last full payment or is instead paid one year after it.
Answer Solve $10000 = 1500\,a_{\overline{n}|8\%}$ , so $a_{\overline{n}|} = 6.666667$ :
 $\begin{align*} n &= \frac{-\ln(1 - 0.08 \times 6.666667)}{\ln 1.08} \\ &= \frac{0.762140}{0.076961} \\ &= 9.90 \end{align*}$ 
So 9 full payments can be made, and something is still owed just after the 9th:
 $\begin{align*} X &= 10000(1.08)^9 - 1500\,s_{\overline{9}|8\%} \\ &= 19990.05 - 18731.34 \\ &= 1258.71 \end{align*}$ 
Balloon: added to the 9th payment, the final payment is $1500 + 1258.71 = \$2{,}758.71$ — 8 payments of $ $1{,}500$ and a 9th, larger one. (The formula above with $n = 9$ gives the same: $10000(1.08)^9 - 1500\,s_{\overline{8}|8\%}(1.08) = 2758.71$ .)

Drop: paid one year later instead, the final payment is $1258.71(1.08) = \$1{,}359.41$ at time 10 — 9 payments of $ $1{,}500$ and a 10th, smaller one. That is a Drop Payment, not a balloon.