Empirical relationship
In science, an empirical relationship is a relationship or correlation based solely on observation rather than theory. An empirical relationship requires only confirmatory data irrespective of theoretical basis. Sometimes theoretical explanations for what were initially empirical relationships are found, in which case the relationships are no longer considered empirical. Thus correlation is not causation but in some cases can be found to result from it. 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 are 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 relationship.
Historically the discovery of empirical relationships has been important as a stepping stone to the discovery of 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.
Usage in physics
In physics, an empirical formula is a mathematical equation that predicts observed results, but is derived from experiment and not directly from first principles.
An example was the Rydberg formula to predict the wavelengths of hydrogen spectral lines. Proposed in 1876, it perfectly predicted the wavelengths of the Lyman series, but lacked a theoretical basis until Niels Bohr produced his Bohr model of the atom in 1925.