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 n1:
If 'n' were used, instead of 'n1', 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 'n1' because only 'n1' of the data values 'x_{i}' are independent; if any 'n1' are known then the other can be calculated (using xbar).
