Many useful problems involve determining whether an input is the member of a class or not, for example,
Given an e-mail, is it spam or not spam?
Given a credit card transaction, is it fraudulent or legitimate?
Given medical test results, are they positive or negative for a disease?
For most such problems, neither a linear nor a polynomial model fits the data very well. One reason is that these models can produce large, continuous-valued, positive or negative outputs. The desire is for a binary output (0 for not-in-class or 1 for in-class).
One recourse is to take the output of a linear or polynomial model and use it as the input to another model with output ranging between 0 and 1. The logistic function has this property: no matter how large or small its input is, the output remains between 0 and 1.
Mathematically speaking, this composes a logistic function with a linear or polynomial function. Data scientists sometimes say the linear/polynomial model’s output is “put through a sigmoid,” or “squashed.” The output of such a model is not strictly binary, but because it is between 0 and 1, we can usefully interpret it as the predicted probability that the input is in a particular class.
The linear regression model assumed that there was a line such that the examples were on the line or near it. Polynomial regression assumed that there was a curve such that the examples were on the curve or near it. Logistic regression assumes that there is a line or curve such that examples on one side of the line or curve are in the class, and examples on the other side are outside the class.