Problem statment: What is the different between standard deviation and variance? What context should it be used for both of them?
standard deviation = square root of variance, hence standard deviation is usually written as $\sigma$ and variance as $\sigma^2$.
Here's a basic explanation comparing the two and how they can be used in basic situations.
In more complicated statistics, variance is used in a lot of ways, such as amount of variance explained (like to calculate $R^2$ or the Intra-Class Correlation coefficient (ICC), which is the amount of variance explained by each level in a multi-level experiment or model.