What is back-emf: counter-electromotive force?

Inductive components like motor winding resist sudden changes in current. That's because the magnetic field caused by the current needs time to build up or decrease. That means that when current is flowing and this is suddenly cut off, the winding will try to maintain that current, and becomes a power source generating a voltage to be able to do so. It gets its power from the built up magnetic field.
Since the winding is now a power source instead of a consumer the voltage is reversed for the same current flow direction. That also explains how the voltage on a coil can become higher than the power supply: instead of subtracting the voltage over it you add it to the power supply. That's why you need a flyback diode on for instance a relay coil: the diode will allow the back emf to flow back to the power supply without damaging the switching transistor.

Motors and generators are somewhat interchangeable things. If you spin a motor, it can generate voltage - even if you spin it by electrical means. Back emf is the voltage produced (generated) in a motor as it spins.

At a dead stop, a motor produces no voltage. If you apply a voltage, and the motor begins to spin, it will act as a generator that will produce a voltage that opposes the external voltage you apply to it. The 'back' part of 'back emf' reflects this opposition.