Polynomial regression is a kind of linear regression.
While linear regression is when both the predictor and the response are each continuous and linearly-related, causing the response to increase or decrease at a constant ratio to the predictor (that is, in a straight line), with polynomial regression, different powers of the predictor are successively added to see if they adjust the response significantly. As these increases are added to the equation, the line of data points will change its shape, turning the linear regression model from a best fitted line into a best fitted curve.
So, why should you bother with polynomial regression? The generally accepted answer or thought process is: when a linear model doesn't seem to be the best model for your data.
There are three main conditions that indicate a linear relationship may not be a good model for a use:
There will be some variable relationships in your data that you assume are curvilinear
During visual inspection of...