What does error refer to in linear regression notation? I regularly see linear regression models written in this notation:
$y = a + \beta X + error$
I've never really pinned down what $error$ actually refers to. In the linear regression plotted below, does $error$ refer to the grey band, therefore the confidence interval for the prediction line?

 A: The model for simple regression is, as you stated, $Y=\alpha+\beta x+\epsilon$ where $\epsilon$ is there error term. In words, that model states that $Y$ is a random quantity composed of a  linear function of a fixed x (that is $\alpha+\beta x$) plus a random error term ($\epsilon$) usually assumed to have mean 0 (and often assumed to be normally distributed). 
In simpler terms, under the linear regression model, the error term explains why all the $y$ values do not lie perfectly on the regression line. If the model were simply $Y=\alpha+\beta x$, the points would all lie exactly on the line. The residuals $e_i=y_i-\hat{y_i}$ are estimates of realizations of the error term for individual realizations of $Y$ and $x$. 
The grey confidence band in you regression plot captures the uncertainty in the estimated regression line. That is, it represents the uncertainty in the estimates of $\alpha$ and $\beta$. This uncertainty is due to the error term, but is not the same as that error term.
A: Another common way do describe the error is the following:
Suppose you have some variable $Y$ which you model as a linear function of another variable $X$, that is, you want to see how well the function $f(X):=\alpha+\beta X$ describes $Y$ according to some measure. Then define $\epsilon$ as the difference between $f(X)$ and $Y$. Then $\epsilon$ is all things affecting $Y$ not captured by $f(X)$. To me, this is the most basic and intuitive way to think about the error term. 
It's everything in $Y$ not explained by a linear (affine) function of $X$.
For example, if you let $Y=$ average age of citizens in a country, and then you model this as an affine function of how much they exercise and how healthy they eat ($X$="Healthy Lifestyle Index"), then $\epsilon$ is everything affecting how long people live on average except for an affine function of some healthy lifestyle index. This includes effects from smoking, the health care system, happiness, climate, etc.
