What is the difference between errors and residuals? While these two ubiquitous terms are often used synonymously, there sometimes seems to be a distinction. Is there indeed a difference, or are they exactly synonymous?
 A: Error term is a theoretical concept that can never be observed, but the residual is a real world value that is calculated for each time a regression is done
A: Errors pertain to the true data generating process (DGP), whereas residuals are what is left over after having estimated your model.  In truth, assumptions like normality, homoscedasticity, and independence apply to the errors of the DGP, not your model's residuals.  (For example, having fit $p+1$ parameters in your model, only $N-(p+1)$ residuals can be independent.)  However, we only have access to the residuals, so that's what we work with.  
A: An error is the difference between the observed value and the true value (very often unobserved, generated by the DGP).
A residual is the difference between the observed value and the predicted value (by the model).
A: Error of the data set is the differences between the observed values and the true / unobserved values. Residual is calculated after running the regression model and is the differences between the observed values and the estimated values.
A: Error term is an unknown value that could never be known unless the DGP is known. Therefore, theoretically, one can generate a variable x from say a Normal random variable and the error from a normal random variable. Then construct the variable $y$ as follow
$$
y_t=\beta x_t+e_t
$$
Here, $e_t$ stands for error term the difference between the true variable $y_t$ and the expected value $\beta x_t$. 
$\beta$ is unknown, once the beta is estimated then we get
$$
y_t=\hat{\beta} x_t+\hat{e}_t
$$
Then $\hat{e}_t$ is not the error anymore it is the residual, the difference between the true value $y_t$ and the estimated value $\hat{\beta} x_t:=\hat{y}_t$. 
This comes in line with another question, what is the difference between the Mean Squared Error and Mean Squared residual. There is nothing called MSR: Means squared residual. 
$$
MSR=\frac{1}{n}\sum_{i=1}^{n}\hat{e}_i
$$
However, many practitioners treat them the same. MSE is a theoretical concept that is always translated to MSR by practitioners due to their unfamiliarity between theory and practice. 
$$
MSE=E(e_{t}^{2})
$$
