Understanding the assumptions of linear regression: residuals or data must come from a normal distribution? I'm trying to better understand linear regression and have always heard that in order to meet the assumption of normality the data has to be normally distributed. However, I've also heard it's more correct to say the residuals should be normally distributed.
Is this because if a data point falls on the linear regression line it is then safe to assume it came from a normal distribution, and in order to make conclusions about the data as a whole, the residuals (the rest of the points that don't fall on the line) must also have come from a normal distribution? But, if the data were all collected together then they must all come from the same distribution, no? Can different parts of the data follow different distributions? Why not just keep saying the data, rather then the residuals, must come from a normal distribution? What is the importance of making this distinction?

First we should distinguish between errors and residuals. If $X_1,\ldots,X_n$ are sampled from a population whose average is $\mu,$ then $X_1-\mu,\ldots,X_n-\mu$ are the errors and $X_1-\overline X,\ldots,X_n-\overline X,$ where $\overline X = (X_1+\cdots+X_n)/n$ is the mean of the sample rather than of the population, are the residuals.
A simple linear regression model says $Y_i=\beta_0+\beta_1 x_i + \varepsilon_i$ for $i=1,\ldots ,n$ and that the errors $\varepsilon_i$ are normally distributed with expected value $0$ are variance $1$ and are mutually indpendent, the parameters $\beta_0,\beta_1$ are not random (and in this context "constant" usually means not random), the parameters $\beta_0,\beta_2$ are not observable but instead must be estimated based on the observed data $(x_i,Y_i),\, i=1,\ldots,n.$
Notice that I have written capital $Y_i$ and lower-case $x_i;$ that is because the $x_i$ are treated as non-random. The word "random" in this context in effect means this: If we toss out the data $(x_i, Y_i),\,i=1,\ldots,n$ and collect new data, that which changes when that is done is "random"; that which does not change is not "random". However, realistically, in many (most) cases, the $x_i$ do change. But that are treated by the model as described above as if they were not random since what is considered is the conditional distribution of $Y_i$ given $x_i.$ (In designed experiments the $x_i$ are in some instances chosen by the experimenter and would remain the same if the experiment is replicated.)
Are the "data" normally distributed? The $Y_i$ are normally distributed because the $\varepsilon_i$ are normally distributed. The errors $\varepsilon_i$ all come from the same distribution; the $Y_i$ come from different distributions because the $x_i$ differ from each other.
But that's just a commonplace and often useful model. Is it "safe to assume" the errors are normally distributed? Certainly not. Statisticians look at the residuals to assess that.
The Gauss–Markov theorem makes weaker assumptions: the errors $\varepsilon_i$ are assumed uncorrelated rather than independent; they are assumed to have expected value $0$ and equal variances but they are not assumed to be identically distributed; hence not assumed to be normally distributed. The theorem states that among all unbiased estimators of $\beta_0,\beta_1$ that are linear as a function of $(Y_1,\ldots,Y_n),$ the one with the smallest variance is the least-squares estimator.
