The emphasis on nonisotropic media in the electromagnetics research community has recently brought forward a large amount of new literature on the material effects. The material phenomena affecting the electromagnetic characterization are contained in the constitutive relations between an electric and a magnetic excitation and an electric and a magnetic response. Starting from the constitutive equations, this article is an attempt to cast light on the labels, terms, notation, and classification of linear electromagnetic materials. Using dyadic analysis and physical concepts like reciprocity and magnetoelectric coupling, the different classes within bi-anisotropic media are presented in systematic form. Simple isotropic media can be characterized by two material parameters: the electric polarizability is measured by permittivity ε, and the magnetic polarizability by the permeability µ. For bi-isotropic media, there exists magnetoelectric coupling, but due to isotropy (independence of the direction of the field vectors) the two additional material parameters are scalars. The physical interpretation to these two parameters are chirality and nonreciprocity. The two subclasses of bi-isotropic materials are Pasteur and Tellegen media. If there is direction dependence in the medium, we call the material anisotropic, and a scalar quantity has to be described by a dyadic with nine components. Finally, the most general material is called bi-anisotropic, which means that in addition to a dyadic permittivity and permeability, the two magnetoelectric material parameters are dyadics. The essential feature in the classification of the present paper is the separation of all the four material parameter dyadics into symmetric and antisymmetric parts. For permittivity and permeability, the symmetric parts correspond to reciprocal media and the antisymmetric parts are nonzero for nonreciprocal media. In the cross-coupling dyadics the decomposition into symmetric and antisymmetric parts disriminates chiral media, omega media, classical magnetoelectric media, and moving media. Finally, possible alternative characterrizations of bi-anisotropic materials are discussed.