How does an outlier affect the value of the standard deviation?

Outliers increase the standard deviation.

Mean is most affected by outliers, since all values in a sample are given the same weight when calculating mean. A value that is far removed from the mean is going to likely skew your results and increase the standard deviation.

Say you have five values: 2, 1, 2, 1.5, and 2.1. Your mean would be 1.72 and your standard deviation would be 0.47. All of your values are pretty close to each other in their distribution, thus your standard deviation is small.

Then say you have five values: 2, 1, 2, 1.5, and 10. Your mean would be 3.3 and your standard deviation would be 3.77. That one outlier (10) makes your standard deviation much greater when compared to the earlier set of numbers. The effect would be less though if our sample size was larger and we only had one outlier.