The only real purpose of the standard errors is as an intermediate value used to compute the confidence intervals. If you want to compare Prism's results to those of other programs, you will want to include standard errors in the output. Otherwise, we suggest that you ask Prism to report the confidence intervals only (choose on the Diagnostics tab). The calculation of the standard errors depends on the sum-of-squares, the spacing of X values, the choice of equation, and the number of replicates.
Prism reports the standard error of each parameter, but some other programs report the same values as 'standard deviations'. Both terms mean the same thing in this context.
When you look at a group of numbers, the standard deviation (SD) and standard error of the mean (SEM) are very different. The SD tells you about the scatter of the data. The SEM tells you about how well you have determined the mean. The SEM can be thought of as "the standard deviation of the mean" -- if you were to repeat the experiment many times, the SEM (of your first experiment) is your best guess for the standard deviation of all the measured means that would result.
When applied to a calculated value, the terms "standard error" and "standard deviation" really mean the same thing. The standard error of a parameter is the expected value of the standard deviation of that parameter if you repeated the experiment many times. Prism (and most programs) calls that value a standard error, but some others call it a standard deviation.