Inference, estimation, and decision-making from data
Fitting a regression is the easy part. The harder question is whether you can trust it. Model diagnostics are the checks that catch a model that fits the numbers but violates the assumptions underneath. The most useful object to look at is the residual: e = y − ŷ, the leftover the model couldn't explain.
If the model is right, the residuals should look like pure noise: no pattern, constant spread, roughly symmetric. The main tool is a residual plot: residuals on the y-axis against the fitted values (or an input) on the x-axis. You're hunting for structure that shouldn't be there.
A good doctor does not stop at naming the illness; they check what symptoms are left over after treatment. If a patient still has a stubborn cough the diagnosis missed something. Residuals are a model's leftover symptoms: the part of the data the fitted line could not explain. If they show a clear pattern instead of harmless random noise, the model has missed something too.