What is the relationship between the standard error of the regression (S) and the standard errors of the coefficients?

The standard error of the regression (S) is an estimate of the standard deviation of the errors. It's related to the standard errors of the coefficients because a larger S (indicating greater error variance) will generally lead to larger standard errors for the coefficients. S is calculated as sqrt(RSS / (n - p)), where n is the number of observations and p is the number of parameters (including the intercept).