what is the difference between 'estimate of residual standard error' and 'residual standard error'? What is the difference between 'estimate of residual standard error' and 'residual standard error'? 
Can someone please provide the formulas?
Thanks!
 A: I assume the residuals you are referring to are from a linear regression model of some sort.
Your first term is easy: the residual standard error (usually just called the standard error) is the MSE (mean square error - you can Google it and easily find the formula, although it is just the square root of the sum of the squared residuals divided by (n-k-1)). Note, that it is a statistic computed from a given data set. 
Generally, in statistics, an "estimate" refers to a statistic that is a "point estimate" for some population parameter (in other words, the single value that  based on our given data set, is our best guess at what the value of some population parameter is). Standard errors are estimates for a particular population parameter, the standard deviation, and in regression specifically, the standard errors is the estimate for the standard deviation of the error term (i.e. $e_i$ is the estimate of $\epsilon_i$). This makes the term "estimate of the residual standard error" a little confusing, since the residual standard error is a statistic and not a population parameter, so to "estimate" it would seem unusual. I wonder if maybe the correct phrase would be the "estimate of the standard deviation for the error term", which the residual standard error is an estimate of.
Without context or reference, not much more I can really say.
