Standard Error of the Mean vs Standard Error of a Proportion

The standard error of the mean measures the spread of the sampling distribution of $\bar{x}$ (x-bar, the sample mean), so it shrinks as the sample grows. In practice you compute $SE_{\bar{x}} = \frac{s}{\sqrt{n}}$ from the sample standard deviation s; if you somehow knew the population standard deviation $\sigma$ (sigma), you would use $\frac{\sigma}{\sqrt{n}}$ instead. For example, with s = 12 and n = 36 the standard error is $12/\sqrt{36} = 12/6 = 2$. Watch the square root: cutting the standard error in half takes four times as many observations.

This is the standard deviation of the sampling distribution of $\hat{p}$ (p-hat, the sample proportion), so it tells you how far a sample proportion typically lands from the true proportion. For a confidence interval the formula is $SE_{\hat{p}} = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$; a significance test puts the null value $p_0$ where p-hat sits, because the test assumes the null hypothesis is true. For example, with p-hat = 0.40 and n = 400, the standard error is $\sqrt{0.40(0.60)/400} = \sqrt{0.0006} = 0.0245$. The standard error is largest when the proportion is near 0.5 and shrinks as it moves toward 0 or 1.