Why N - 1?

The n-1 equation is used in the common situation where you are analyzing a sample of data and wish to make more general conclusions. The SD computed this way (with n-1 in the denominator) is your best guess for the value of the SD in the overall population.

Why do we use n-1 in standard deviation?

The intuitive reason for the n−1 is that the n deviations in the calculation of the standard deviation are not independent. There is one constraint which is that the sum of the deviations is zero.

Why do we subtract 1 from n?

So why do we subtract 1 when using these formulas? The simple answer: the calculations for both the sample standard deviation and the sample variance both contain a little bias (that's the statistics way of saying “error”). Bessel's correction (i.e. subtracting 1 from your sample size) corrects this bias.