Running tally
What you just did has a name
A regression line is a summary. It reduces a two-variable scatter to a slope and intercept — and throws away everything else. The residuals (observed − fitted) are what gets thrown away. If they look like clean noise, the summary was faithful. If they show structure — curvature, clusters, changing spread — then the model missed something, and that something can be identified by looking at the residuals.
Four patterns cover most cases in practice: clean, curvature, two clouds, heteroskedasticity. You just saw all four. Curvature says "this isn't a line — try a log transform or a nonlinear model." Two clouds says "there's a grouping variable you forgot to include — species, guild, sex, treatment." Heteroskedasticity says "variance depends on x or on the mean — weighted least squares or a log transform will help."
This is the opening move of every real data analysis. Clean residuals do not confirm the model — dirty residuals reject it. The residual plot is cheaper than the p-value; look at it first.