A transformation of a parameter governing a response distribution that is used as a crucial part of the generalized linear model to map that parameter's range (which may be from 0 to 1, or only positive values, e.g.) to the real number line $(-\infty, +\infty)$.

Link functions are a central part of the Generalized Linear Model. Many non-normal response distributions (e.g., binomial, Poisson, etc.) are governed by parameters that can only range over a bounded interval. For example, the mean of the Bernoulli distribution is $\pi$, the probability of 'success', which can only range from 0 to 1. However, the structural part of a model, $\beta_0 + \beta_1X$, can range from $(-\infty, +\infty)$. The link function allows the predicted parameter to be equated to the structural part by transforming the parameter such that the transformed parameter can range from $(-\infty, +\infty)$.

Wikipedia https://en.wikipedia.org/wiki/Generalized_linear_model#Link_function has more information and references