In science, an empirical relationship is one based on observation rather than theory: that is, there is no theoretical reason to believe that a relationship should be as claimed; only data that indicates it is. Sometimes theoretical explanations for what were initially empirical relationships are found, in which case the relationships are no longer considered empirical. Other times the empirical relationships are merely approximations, often equivalent to the first few terms of the Taylor series of the "real" answer (though in practice these approximations may be so accurate it is difficult to tell they're approximations). Still other times the relationships may later be found to only hold under certain specific conditions, reducing them to special cases of more general relationships.
Historically the discovery of empirical relationships has been important as a stepping stone to the discovery of true theoretical relationships. And on occasion, what was thought to be an empirical factor is later deemed to be a fundamental physical constant.
An empirical equation is simply a mathematical statement of one or more empirical relationships in the form of an equation.