Suppose that we have data points (X_1, y_1), \dots, (X_n, y_n); we operate under the framework that X_i are always implicitly conditioned on. The generalized linear model will allow us to generalize from continuous real values to more diverse outputs. Particularly common is y_i arising from an exponential family.
Def: A link function is how E[y_i] (or E[y_i \mid X_i]) depends on X_i. In particular, we will have g(E[y_i]) = g(\mu_i) = X_i^T \beta.