Quantum entanglement is an astonishing phenomenon of quantum mechanics, which is treated in this paper with a theoretical approach. Quantum entanglement denotes the appearance of a connection between two measurement outcomes: the measurement outcome on one component may be completely indeterminate, but the measurement result completely determines the state of the other component. Quantum entanglement is studied in composite systems. The composite system in this paper is a system of two spins \(\sigma\) and \(\tau\), and, each with its own three x, y and z components. Examples of a composite system are a product state, where the two spins are prepared independently, and which shows no correlation between measurements and no coupling. On the other hand, the singlet state is an example of a composite system that shows maximum coupling. The characteristic of maximum coupling is that, although the characteristics of the system as a whole are known, it is shown that there is no information about subsystems as individual elements. The paper also studies the effect of the operator on the product state and the singlet state. Finally, quantum entanglement entails the EPR paradox, a thought experiment that questions the completeness of quantum mechanics. The solution to the EPR paradox is Bell's theorem which shows that quantum mechanics is a nonlocal theory.