To complete the answer to the question, Ocram nicely addressed standard error but did not contrast it to standard deviation and did not mention the dependence on sample size. As a special case for the estimator consider the sample mean. The standard error for the mean is $\sigma \, / \, \sqrt{n}$ where $\sigma$ is the population standard deviation. So in this example we see explicitly how the standard error decreases with increasing sample size. The standard deviation is most often used to refer to the individual observations. So standard deviation describes the variability of the individual observations while standard error shows the variability of the estimator. Good estimators are consistent which means that they converge to the true parameter value. When their standard error decreases to 0 as the sample size increases the estimators are consistent which in most cases happens because the standard error goes to 0 as we see explicitly with the sample mean.

Incidentally, this exact question came up with a colleague today.

If you want to do the z-score, you use standard deviation: $z = \frac{x - \bar{x}}{s}$. Standard deviation is a property of a population, and we use some formula (often $s^2 = \frac{1}{n-1}\sum_{i=1}^n (x_i - \bar{x})^2)$ to estimate the population parameter. The population parameter does not depend on the sample size or even the sample at all.

Standard error has to do with the so-called sampling distribution of a statistic. This does depend on the sample size. Standard error of the mean is what most people see first. You know the formula: $\dfrac{s}{\sqrt{n}}$.

The sampling distribution of a statistic follows the following logic.

1) Pick of statistic of interest, say $\bar{x}$.

2) Grab a sample of size n.

3) Calculate $\bar{x}$.

4) Repeat 2 and 3 (infinitely) many times.

5) Plot the values of $\bar{x}$ that you calculated (something like a histogram). to get the sampling distribution.

Since you should have tighter and tighter estimates of $\mu$ as you get larger and larger sample sizes, your $\bar{x}$ values should be very close to $\mu$ when you have a lot of data, and this is why standard error shrinks as you increase the sample size. However, if we sampled from a standard normal population, no matter how big the sample size (and correspondingly small the standard error), the population has a standard deviation of 1.

Since students first learn about standard error of the mean, which explicitly depends on the population (or estimated) standard deviation, and students often do not progress past testing means, it's easy to confuse the two. (In fact, the standard error is the standard deviation of the sampling distribution.) However, you can find a standard error for other statistics. Look at those steps 1-5. You could repeat that process for a different statistic, perhaps $s^2$, median, or IQR. If you know how to write loops in a language like Python or R (or whatever statistical language), I'd encourage you to try writing a simulation of these.