Why the variance formula divides by n minus 1

A bottling line fills $n$ bottles and you record the volumes $X_1, \dots, X_n$, an i.i.d. sample with unknown mean $\mu$ and unknown variance $\sigma^2$. You estimate the process variance by averaging the squared gaps from the sample mean $\bar X$.

If you want the estimator to be correct on average (unbiased) for $\sigma^2$, what number should you divide $\sum_i (X_i - \bar X)^2$ by, and why is it not $n$?