back emf:
when the motor is running, the coil constantly cuts the magnetic field thus there is a changing magnetic field, by faraday, this cutting flux would induce a current, and by lenz's law, this induced current would oppose the motion of the cause, thus this current is in the opposite direction of the emf input, and so it's called back emf.
eddy current:
still using the idea of lenz's law, when a moving conductor(in this case a piece of metal or watever) is in a magnetic filed, and changing the strength of the magnetic filed, it experiences a force, which also has currents induced in it. in this case the current induced is circular, and the force is trying to act against the motion of the metal piece or something. so u can work out whether the curent is anticlockwose or clockwise using hand rules .