Considering composite quantum systems one can ask whether the system behaves in a classical or quantum way. This is related to the type of correlations between their subsystems. I will start my presentation with introducing the classification of composite quantum systems and in the following parts I will elaborate mainly on the quantum correlations. I will explain the notion of quantum discord as a measurement induced informational measure of quantum correlations and discuss it's properites. Moreover, methods for quantum discord investigation in bipartite two-level systems will be presented along with the difficulties arising when moving to higher level systems.