The DC motor is supplied with DC power but the coil inside the rotor is still in the AC circuit. DC motors with brushes are commutated by brushes. And brushless motors change DC to AC through an inverter circuit and then **** to the coil. Commonly used in 12V cooling fans and such. Commonly used in computer mainframes.
AC motors do not need commutator (brushes). Generally do not need permanent magnets. ...... Want to know more about it buy a book on the principles of motors and just read it