Back-emf is the voltage that a motor produces as it is spinning - the generator action of the motor.
Flyback voltage is a large voltage spike that occurs when you quickly switch off a motor coil (or any inductor). These spikes can be thousands of volts. So they must be snubbed with diodes or risk destroying the switching electronics. Fast-recovery diodes need to be used because the diode is reversed biased when the coil is energized, but must quickly switch to forward conducting to snub the flyback voltage when the coil is de-energized.