In a brushed motor, brushes (similar to paint brushes or more comonly soft metal bars.) contact the armature and allow electricity to flow into it to create a magnetic field. The magnetic field of the amrature opposes the magnetic field generated by the stator windings surrounding the armature. This is what causes the motor to spin. (Basically) As the armature spins, the brushes contact a different set of windings on the armature altering the magnetic field in it so it continually opposes the stator winding fields.
In a brushless motor, the armature windings are replace with permanent magnet and the roles of the armature and stator are reversed. The brushless motor requires an electronic controller to determine where the armature is in it's rotation with respect to the stator windings so it can alter the magnetic field of the stator windings to keep the armature spining.
A brushless motor is more costly to make, usually more powerful, lasts longer,but simpler mechanically to make.