A linear regression model is a statistical model for linear relationships between variables. More precisely, it is a model for how the expected value (mean) of a response variable changes linearly with the value of one, or more, explanatory variables. It is useful for description, prediction or to improve understanding of a physical process.
As with all statistical models, it involves assumptions about the stochastic (or random) behaviour of the variables that we are modelling. Two models are often discussed in text books: simple linear regression and multiple linear regression. As we shall see, the former is a special case of the latter.