The present thesis is devoted to studies of entanglement properties of pure twoqubit, qubit-qutrit and two-qutrit states. Entanglement is essentially non-classical characteristics of composite quantum states and it plays a key point in quantum computation and quantum information theory. To characterize entanglement of the states we use the reduced density matrix approach, which relates entanglement of pure composite state with mixed reduced density matrix. Von Neumann entropy of the reduced density matrix and the linear entropy, as squared concurrence are used to quantify entanglement. By using the unitary one qubit and one qutrit gates we show invariance of entanglement under the transformations. This allows us to construct continuously parametrized set of the same level entangled states. We apply the results for calculation of purification for given mixed state and for finding maximally entangled minimum of average energy for two-qubit spin XYZ model in magnetic field.