Residuals

Gist

The vertical distance between the predicted and observed values. Supposed to be the proxy of Error

Linear regression

In Linear regression, the residuals are calculated as: $r_{i} = y_{i} - M o d e l O u t p u t$ .

Residual standard deviation

Summarizes the scale of the residuals

Other uses

Some people use the residuals from a regression as data but that seems to be a bad idea most of the time (Gary King 1986, Freckleton)

Diagnostic plots (Linear Regression)

Reference

Gelman, A., & Hill, J. (2007). Data analysis using regression and multilevel/hierarchical models. Cambridge university press.
Freckleton, R. P. (2002). On the misuse of residuals in ecology: regression of residuals vs. multiple regression. Journal of Animal Ecology, 542-545.
Gary King. 1986. “How Not to Lie With Statistics: Avoiding Common Mistakes in Quantitative Political Science”. American Journal of Political Science, 30, Pp. 666–687.