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The number of Degrees of Freedom is the number of independent data values that are used in estimating the value of a population parameter.
The number of degrees of freedom in the standard deviation formula is n-1:
If 'n' were used, instead of 'n-1', the value of 's' would be biased; the standard deviation calculated from small samples would underestimate the population standard deviation.
The number of degrees of freedom is 'n-1' because only 'n-1' of the data values 'xi' are independent; if any 'n-1' are known then the other can be calculated (using x-bar).
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