Transistors do most of the work inside of amps; they are what create a stronger electrical signal to send to the speaker.  Basic transistors consist of three layers of semiconductors and three nodes; the base, emitter and collector nodes.  Two negatively charged layers of semiconductors sandwich a layer of positively charged semiconductor.  When there is no signal from the guitar, the electrons fill in the positively charged gaps between the layers.  This creates a neutral area preventing the negatively charged electrons from passing through.  When current is sent to the base conductor from the guitar, the electrons flow into the transistor scattering the electrons that had filled the gaps.  These gaps allow the electrons to flow from the power source, through the layers, and to the speaker recreating the sound.  If the string on the guitar is plucked hard, it will send a strong signal to the transistor recreating the sound louder than if it was plucked softly.  When a strong signal is sent from the guitar then a lot of the electrons scatter inside the layers allowing a strong signal to be sent to the speaker.  If a weak signal is sent then less of the electrons scatter inside the layers allowing a weaker signal to the speaker. 

The next part inside the amp is the speaker.  The speaker receives the signal from the transistor (from the collector node) and turns it back into sound waves.  Speakers work in the opposite way that the pick-ups do.  The speaker uses a magnet and the electronic signals to recreate the sound waves.

The signal flows through the green wires in the picture shown below.  As a signal is sent it creates a negative charge around the core pushing it away from the magnet.  As the core is pushed away from the magnet it also pushes the cone.  The cone is made out of a light, flexible material so when the core pushes up the cone reacts in the same way.  As a series of signals are sent to the speaker it causes the cone to vibrate causing a sound wave to radiate from the speaker.  As the sound waves leave the speaker they should arrive at your ears shortly, then you can enjoy the musical presentation!