This paper studies a class of systems with discontinuous right-hand side, which is commonly used in various applications. The notion of discontinuous system is closely linked to the notion of differential inclusion, which was first considered by Marchaud and Zaremba. In this paper three different notions of solutions of differential equations will be considered: Filippov, Aizerman-Pyatnitskiy and Gelig-Leonov-Yakubovich solutions. For the class of systems considered in the paper it is discussed when these definitions coincide and when they differ. The application of definitions is also demonstrated by numerical modelling of hidden attractor in Chua's circuit.