Why do we say "Residual standard error"? A standard error is the estimated standard deviation $\hat \sigma(\hat\theta)$ of an estimator $\hat\theta$ for a parameter $\theta$.
Why is the estimated standard deviation of the residuals called "residual standard error" (e.g., in the output of R's summary.lm function) and not "residual standard deviation"? What parameter estimate do we equip with a standard error here? 
Do we consider each residual as an estimator for "its" error term and estimate the "pooled" standard error of all these estimators?
 A: From my econometrics training, it is called "residual standard error" because it is an estimate of the actual "residual standard deviation". See this related question that corroborates this terminology.
A Google search for the term residual standard error also shows up a lot of hits, so it is by no means an R oddity. I tried both terms with quotes, and both show up roughly 60,000 times.
A: I think that phrasing is specific to R's summary.lm() output.  Notice that the underlying value is actually called "sigma" (summary.lm()$sigma).  I don't think other software necessarily uses that name for the standard deviation of the residuals.  In addition, the phrasing 'residual standard deviation' is common in textbooks, for instance. I don't know how that came to be the phrasing used in R's summary.lm() output, but I always thought it was weird.
A: 
Put simply, the standard error of the sample 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.  

Standard error - Wikipedia, the free encyclopedia
A: A fitted regression model uses the parameters to generate point estimate predictions which are the means of observed responses if you were to replicate the study with the same XX values an infinite number of times (when the linear model is true).
The difference between these predicted values and the ones used to fit the model are called "Residuals" which, when replicating the data collection process, have properties of random variables with 0 means. The observed residuals are then used to subsequently estimate the variability in these values and to estimate the sampling distribution of the parameters. 
Note:
When the residual standard error is exactly 0 then the model fits the data perfectly (likely due to overfitting). 
If the residual standard error can not be shown to be significantly different from the variability in the unconditional response, then there is little evidence to suggest the linear model has any predictive ability.
A: For the nls (nonlinear least squares fit) R function, the "Residual standard error" seems to be:
$$
\sqrt{\frac{\mathrm{RSS}{n-p}}
$$
where RSS is the "residual sum-of-squares", n is the number of observations and p is the number of estimated parameters. There's absolutely no description in the documentation, this assumption is based on a "numerical experiment".
