Intuitive Explanation of Bessel's Correction When calculating a sample variance a factor of $N-1$ appears instead of $N$ (see this link ).  Does anybody have an intuitive way of explaining this to students who need to use this fact but maybe haven't taken a statistics course?

The sum $\sum (x_i - a)^2$ is minimized when $a$ is the average of the $x_i$'s.  The proof is a simple exercise in algebra (write sum as quadratic in $a$) or calculus (differentiate to get $\sum (x_i - a) = 0 $).
Therefore, $\sum (x_i - \overline{x})^2 \leq \sum (x_i - \mu)^2$.
The same inequality holds with averages in place of sums.  
The average of the squared $(x_i - \mu)$'s is an unbiased estimate of the variance, since each term has that variance as its expected value.  Replacing $\mu$ by $\overline{x}$ produces a smaller estimate of the variance, and the expected value of that estimate is lower than the expected value of the unbiased estimator with $\mu$.  The latter is the variance, so the expression with $1/N$ will (on average) underestimate the variance. 
To summarize: the use of $\overline{x}$ as a surrogate for $\mu$ causes a downward bias in estimating variance by $\sum (x_i - \overline{x})^2 / N$.
