A linear regression is a regression that is linear in the unknown parameters used in the fit.
The most common form of linear regression is least squares fitting. Least squares fitting of lines and polynomials are both forms of linear regression.
The linearity assumption is that the conditional expectation value of the response has the stated linear form in the model parameters. It does not by itself require the response to be a linear function of every raw predictor, since transformed predictors and polynomial terms can still enter linearly through their coefficients. Assumptions such as independent errors, constant variance, and a normal distribution are additional conditions used for particular estimators or inferential procedures, not part of the definition of linear regression.
