What are linear models?

In building a statistical model for  experimental or observational data we often want to characterise the dependence of a response, such as a patients heart rate, on one or more covariates, e.g. treatment group or control group. We also want to characterise the "unexplained" variation in the response. Empirical models (I.e. models that are derived from the data itself) are generally chosen to be linear in the parameters as these are much simpler to use than are nonlinear models. A covariate can be repeatable (e.g treatment) or non repeatable (e.g. patient). Our goals with modeling repeatable covariates and non-repeatable covariates are different. With a repeatable covariate we want to characterise the change in the response between different levels and for this we use fixed-effects terms that represent, say, the typical rate of change of the response with respect to time under treatment. For a non-repeatable covariate we want to characterise the variation induced in the response by the different levels of the covariate and for this we use random-effects terms. A statistical model that  incorporates both fixed-effects terms and random-effects terms is called a mixed-effects model, a mixed model.
See R News may 2005