This tutorial provides an intuitive and concrete description of the phenomena of electromagnetic nonreciprocity that will be useful for readers with engineering or physics backgrounds. The notion of time reversal and its different definitions are discussed with special emphasis on its relationship with the reciprocity concept. Starting from the Onsager reciprocal relations that are generally applicable to many physical processes, we present the derivation of the Lorentz theorem and discuss other implications of reciprocity for electromagnetic systems. Next, we identify all possible routes toward engineering nonreciprocal devices and analyze three of them in detail, based on external bias and on nonlinear and time-variant systems. The principles of the operation of different nonreciprocal devices are explained. We address the similarity and fundamental difference between nonreciprocal effects and asymmetric transmission in reciprocal systems. In addition to the tutorial description of the topic, this article also contains the original findings. In particular, the general classification of reciprocal and nonreciprocal phenomena in linear bianisotropic media based on the space- and time-reversal symmetries is presented. This classification serves as a powerful tool for drawing analogies between seemingly distinct effects having the same physical origin and can be used for predicting novel electromagnetic phenomena. Furthermore, electromagnetic reciprocity theorem for time-varying systems is derived, and its applicability is discussed.