Residual plots graph the distance of each data point from the curve in a chosen model, and can be used to tell if a given data set fits a selected model. A residual is the distance of a point from the curve. When making a residual plot, the x-axis is the same as in the graph of the data, and the y-axis is the residual, or the distance of a point from the curve. Residuals are positive when the point falls above the curve and negative when it falls below it.
Residual plots can be used to determine if the data fit a given model. If the points on a residual plot are scattered randomly about the x-axis, then the model is a good fit for the data. Otherwise, another model might be more optimal. Points should be distributed randomly around the x-axis because for any set of instances, the error between the model and the observation should be random. You shouldn’t be able to predict the error for any specific observation. This gives a good mathematical explanation of why residual plots should be used to check your analysis.
For example, in the image shown here, the residual plots show that the 2nd order regression fits the data better than the linear regression because the data are more randomly distributed across the x-axis in the residual plot for the 2nd order model.
Here you can find examples of residual plots.
A good summary of some of the type of information we can deduce from residual plots can be found here.