I'm having a hard time verbalizing the difference between these two concepts. How does this sound?
Standard deviation: perform a measurement; there is a 68% chance that that measurement will be within one standard deviation of the sample mean. Qualitatively, measures scatter/variation/dispersion of the data. It does not change with sample size (assuming large N).
Standard deviation of the mean (or standard error): perform a series of measurements; there is a 68% chance that that sample mean (or average) will be within one “standard deviation of the mean” of the population mean. Decreases with increasing sample size.