Back emf is the emf induced in the coils of a motor as they spin in the magnetic field of the stator.
By Lenz's Law, the direction of that induced emf opposes the emf causing the motion of the armature. The current generated in the motor is an eddy current. The direction of the motor eddy current is such that it opposes the supply emf that produces the motion in the motor. The net emf applied to the coils equals the supply emf minus the back emf.
As the coil rotates faster, the back emf increases and the difference between the constant supplied emf and the back emf gets smaller. Clearly, this difference between the two emfs is equal to the potential difference across the motor coil and hence determines the actual current in the coil.
(This part's about the burning out bit)
When the motor is first turned on and the coil begins to rotate, the back emf is very small, since the rate of cutting flux is small. This means that the current passing through the the coil in the forward direction is very large and could possibly burn out the motor.
To ensure that this doesnt not happen, adjustable starting resistors in series with the motor are used, especially with large motors.
Once the motor has reached its normal speed, these starting resistors can be switched off, since by then the back emf has reached a maximum and has therefby minimised the current in the coil.
Also, if the load on the motor has increased, the motor will slow down reducing the back emf and allowing a larger current to flow in the coil. Since torque is proportional to current, an automatic increase in torque will follow an increase in load on the motor, possibly burning it out.