A linear regression model aims at explaining the behavior of one variable with another one, or several others, and by so doing, the assumption is that the relationship between the variables is linear. In general, the expectation of the target variable, the one you need to explain, is an affine transform of several other variables.
Linear models are presumably the most used statistical models, mainly because of their simplicity and the fact they have been studied for decades, leading to all possible extensions and analysis one can imagine. Basically all statistical packages, languages, or software implement linear regression models.
The idea of the model is really simple: a variable y is to be explained by several other variables xi by assuming a linear combination of x's—that is, a weighted sum of x's.
This model appeared in the 18th century in the work of Roger Joseph Boscovich. Then again, his method has been used by Pierre-Simon de Laplace, Adrien...