First, a basic understanding of a few properties of electromagnetism:

• Magnetic fields apply forces to moving charges (electrons).

       Induced Current

• Charges moving through a magnetic field cause a change in the magnetic field.  When a conductive rod moves through a magnetic field, the same effect is exhibited.
• Moving a coil (of wire) through a magnetic field induces a current in the coil.  In other words, a change in the magnetic field around a coil causes a current in the coil.
• The frequency of such a change is proportional to the frequency at which current is induced. 
  

       Magnetic Flux

• Change in a magnetic field.  Lenz's Law:   Flux = BAcos (theta)   where B is strength of a magnetic field.
  

                                                                                                                             A is area of the imaginary plane a rod forms by its motion.
                                                                                                                             (  phi  ) is the angle from the normal of A to the magnetic field.      

The Induced Electromotive Force (emf) in a loop,  the force causing a current,   emf = (delta Flux) / (delta t)     where t is time in seconds.

Now, applying these concepts to the electric guitar:

1. Each string sits above a pickup, which is a magnet with a coil wrapped around it  (approximately 7,000 times). 
   
2. Vibrations in a string cause a change in flux, mostly by creating and changing an area.
3. The change in flux creates an emf, causing current to flow from the pickup to the external amplifier.
4. The amp boosts the signal from the guitar through a circuit of transistors, resistors, and capacitors, and puts out enough current to vibrate the diaphragm of a speaker at the original frequency of the vibrating string.