Y-Hat vs Residual

The predicted value is the height of the regression line above a chosen $x$, not a value anyone observed. You get it by substituting: with $\hat{y} = 12 + 3x$ (read y-hat), an $x$ of 4 gives $\hat{y} = 12 + 3(4) = 24$. If the individual with $x = 4$ actually had $y = 26$, the residual is $26 - 24 = 2$, so the model underpredicted by 2. Predicted values stay trustworthy only inside the range of $x$ used to fit the line; outside that range you are extrapolating.

A residual measures how far off the line is for a single point, so a positive residual sits above the line and a negative one below. It equals $\text{residual} = y - \hat{y}$, the observed response minus the predicted response $\hat{y}$ (read y-hat). For example, if the line predicts 72 but a student actually scored 80, the residual is $80 - 72 = 8$. Least-squares regression chooses the line that makes the sum of the squared residuals as small as possible.