We study the nonequilibrium spin transport through a quantum dot coupled to the magnetic electrodes. A formula for the spin-dependent current is obtained and is applied to discuss the linear conductance and magnetoresistance in the interacting regime. We show that the Kondo resonance and the correlation-induced spin splitting of the dot levels may be systematically controlled by internal magnetization in the electrodes. As a result, when the electrodes are in parallel magnetic configuration, the linear conductance is characterized by two spin-resolved peaks. Furthermore, the presence of the spin-flip process in the dot splits the Kondo resonance into three peaks.