Naturally, I would expect my model to be unbiased, at least in intention, and hence any leftovers on either side of the regression line that did not make it on the line are expected to be random, i.e. without any particular pattern.
That is, I expect my residual error distributions to follow a bland, normal distribution.
In R, you can do this elegantly with just two lines of code.
1. Plot a histogram of residuals
2. Add a Quantile-Quantile plot with a line that passes through, namely, the first and third quantiles.
There are several more techniques in here to analyze residuals, so check it out.