What situation calls for dividing the standard deviation by $\sqrt n$?

This formula may be derived from what we know about the variance of a sum of independent random variables.[4]

If $X_1, X_2 , \ldots, X_n$ are $n$ independent observations from a population that has a mean $\mu$ and standard deviation $\sigma$, then the variance of the total $T = (X_1 + X_2 + \cdots + X_n)$ is $n\sigma^2$.

The variance of $T/n$ must be $\frac{1}{n^2}n\sigma^2=\frac{\sigma^2}{n}$. And the standard deviation of $T/n$ must be $\sigma/{\sqrt{n}}$. Of course, $T/n$ is the sample mean $\bar{x}$.

More explanation @http://en.wikipedia.org/wiki/Standard_error

In the normal distribution, if the expectation of the average of a sample size n is the same as the expectation, however, the standard deviation of your sample is to be divided by the square root of your sample size. You may read about Square Root n Law or Central Limit theorem, which should be in your stats book somewhere.

The answers here don't mention a couple key points. So just wanted to clarify based on my understanding.

This formula is used to calculate the standard deviation of a sample distribution of the mean (of a large number of samples from a population). In other words, it's only applicable when you are looking for the standard deviation of means each calculated from a sample of size $n$, taken from a population.

So lets say i take a population and sample it $10\,000$ times with a sample size of $n=2$. I then take the mean of each one of those samples (so my data contains $10\,000$ calculated means). This equation provides that with a large enough number of samples, the standard deviation of the sample means can be approximated using this formula:

$\frac{\sigma}{\sqrt{n}}$ or in this case $\frac{\sigma}{\sqrt{2}}$

It should be intuitive that as $n \to p$ (population size), your standard deviation of your sample means will become very small, or in other words the means from each sample will have very little variance (if every sample out of $10\,000$ you sampled the entire population you would get no variance from the population mean).

With certain inference conditions (our sample is random, normal, independent) we can actually use this standard deviation calculation to estimate the standard deviation of our population. Since this is just an estimate, its called the Standard Error. The condition for using this as an estimate is that your sample size n is greater than 30 (given by the central limit theorem) and meets the independence condition n <= 10% of population size.