The standard deviation measures variability or dispersion of a sample. (Of course, the standard deviation is also used to refer to a parameter of a distribution or a characteristic of a population).
The standard error measures the precision of an estimate.
The standard error of a statistic (usually an estimate of a population parameter) is an estimate of the standard deviation of its sampling distribution.
In particular, the standard error of the mean is the standard deviation divided by the square root of the sample size.
The standard deviation of a sample is a descriptive statistic, whereas the standard error of the mean is descriptive of the random sampling process. The standard deviation of the sample data is a description of the variation in measurements, while the standard error of the mean is a probabilistic statement about how the sample size will provide a better bound on estimates of the population mean, in light of the central limit theorem.
Put simply, the standard error of the sample mean is an estimate of how far the sample mean is likely to be from the population mean, whereas the standard deviation of the sample is the degree to which individuals within the sample differ from the sample mean. If the population standard deviation is finite, the standard error of the mean of the sample will tend to zero with increasing sample size, because the estimate of the population mean will improve, while the standard deviation of the sample will tend to approximate the population standard deviation as the sample size increases.
In practice, if you want to show the precision of an estimate, the 95% confidence interval is often preferred than the standard error (which is, in fact, a sort of 67% confidence interval).
As a side note, estimators have a meaning that is different from estimates.
An estimator is a rule for combining data to produce a numerical value for a population parameter; the form of the rule does not depend on the particular sample obtained.
An estimate is the numerical value taken on by an estimator for a particular sample of data.
An estimator is a random variable, while an estimate is a realization of an estimator based on a sample.
A statistic is a summary of a sample: it is any quantity computed from values in a sample,
We can use the sample mean as the estimator of the population mean. The sample mean (take the sum of all the observation and divide it by the sample size) is an estimator, and the sample mean of a specific sample (which is a statistic of the sample) is an estimate of the population mean.
