AC synchronous motors run at the same frequency as the electric power supply while asynchronous motors do not neccessarily run at the same speed (frequency), that is their speed usually vary as the load on them changes.
With DC motors the out magnets remain stationary while the inner rotor consists of electromagnets that are powered via commutators from the DC power supply. The polarity of the inner rotor changes as it spins around. It is first attracted towards the outer magnets and then repelled as it passes by only to be attracted towards the next outer magnetic pole.
With AC motors the situation is usually reversed - or one finds a combination of changing magnetic fields in the stator and rotor - with both consisting of electromagnets.