Residuals
A fitted line got some points right and missed others. What do the misses say about the line, and about whether a line was the right choice at all?
Where you are 3 inside
What this area is about
Every point in a regression has a residual: actual minus predicted, y - \hat{y}. A positive residual means the point sat above the line, a negative one below. One residual describes one individual. All of them together describe the line: whether its shape was right, and how far off its predictions typically are.
How these fit together
The procedure for one point. Interpreting a residual reads a single miss in context: how far, on which side, and in what units.
The procedure for all of them. Residual plots graph each residual against the explanatory variable. A shapeless band says a line was a fair choice; a curve says it was the wrong shape, and a fan says its predictions grow less sure as x grows.
Two summaries. r^2 and the standard deviation of the residuals turns the misses into two summaries: the share of the variation in the response the line accounts for, and the typical size of a prediction error.
A path through it
Interpret one residual first, then make residual plots, and finish with r^2 and the standard deviation of the residuals.
This is where regression inference starts: the conditions for inference are checked on residuals, regression output reports their summaries, and a curved residual plot is the cue for transforming to linearity.