The purpose of Regression analysis is to develop a cause and effect "Model" in the form of an equation. To keep things simple, let's talk about Simple Linear Regression, in which the equation is y = bx + a. The Regression analysis comes up with the values for b and a.
Residuals represent the Error in the Regression Model. They represent the Variation in the y variable which is not explained by the Regression Model.
So, Residuals must be Random. If not -- if Residuals form a pattern -- that is evidence that one or more additional Factors (x's) influence y.
A Scatterplot of Residuals against y-values should illustrate Randomness:
Being Random means that the Residuals
Here are a couple of patterns which indicate the Regression Model is incomplete.
Andrew A. (Andy) Jawlik is the author of the book, Statistics from A to Z -- Confusing Concepts Clarified, published by Wiley.