Let me jot down an example to help you visualize residuals:
Those bold blue lines are the residuals of our models. How do we compute an index that is able to summarize how our model performs overall? We could just sum up all our errors and obtain a comprehensive measure of our errors. But, what about underestimation and overestimation?
Let's take, for instance the following model:
As you can see, on the first and third point the model overestimates the actual value of y, while the model underestimates the actual value of ...