

The standard error of the mean (SEM) quantifies the precision of the mean. It is a measure of how far your sample mean is likely to be from the true population mean. It is expressed in the same units as the data.
GraphPad Prism uses the abbreviation SEM, but some prefer (insist on) the abbreviation SE (1, 2).
The SEM is always smaller than the SD. With large samples, the SEM is much smaller than the SD.
Although scientists often present data as mean and SEM, interpreting what the SEM means is not straightforward. It is much easier to interpret the 95% confidence interval, which is calculated from the SEM.
With large samples (say greater than ten), you can use these rulesofthumb:
The 67% confidence interval extends approximately one SEM in each direction from the mean.
The 95% confidence interval extends approximately two SEMs from the mean in each direction.
The multipliers are not actually 1.0 and 2.0, but rather are values that come from the t distribution and depend on sample size. With small samples, and certainly when N is less than ten, those rules of thumb are not very accurate.
No. Statistical computations can compute a standard error for almost any parameter computed from a sample of data. Prism can compute the standard error of a slope in linear regression, and any parameter (i.e. rate constants) from nonlinear regression. The abbreviation SE applies to any standard error, including the standard error of the mean in many journals. The abbreviation SEM always applies to the standard error of the mean.
1.CurranEverett D, Benos D. Guidelines for reporting statistics in journals published by the American Physiological Society. AJP  Gastrointestinal and Liver Physiology. 2004 Aug 1;287(2):G307.
2.Ludbrook J. The presentation of statistics in Clinical and Experimental Pharmacology and Physiology. Clin Exp Pharmacol Physiol. 2008 Oct 1;35(10):1271–4; authorreply1274.