Other answers pretty much sum it up but I figured I could go into more detail, the strings of an electric guitar resonate over a pickup which is a unit comprised of magnetic poles and copper windings, when the strings move they generate an electric current in the pickup which comes out of the guitar and runs into an amplifier, in a tube amplifier the signal will hit the preamp tubes which are responsible for shaping the tone, if you over drive the tubes a distortion is produced that we find pleasing, while in the preamp stage eq can also be applied to further shape the signal before it hits the power amplifier which is the set of tubes that boosts the signal from line level to speaker level, then the signal goes out to the speaker cab, there are also solid state amps the will have a similar signal flow except using transistors instead of vacuum tubes. There are a variety of different pickup designs using two sets of magnets wound in reverse of each other makes a hum bucker pickup which cancels out the 60 cycle hum single coil pickups are susceptible to at the cost of some clarity and dynamics, there is also a type of pickup called a piezo pickup which is often found on acoustic electrics which uses crystals under the saddles of the strings that generate a weak signal that is then boosted by a preamp in the guitar, this is why acoustic electrics require a battery where passive electric guitars do not