As an audiophile I thought this could be a fun one
Dynamic Drivers -
Also known as moving coil drivers, these headphones use a static magnetic field and a moving diaphragm to create sound. Inside is a stationary magnetic element surrounded by a light weight, stiff diaphragm used to create the sound waves. The diaphragm is attached to a coil of wire which allows a current to create the alternating magnetic field required to move the diaphragm up and down along the stationary magnet to create sound waves.
Unfortunately I could not find the picture I wanted to use to explain these drivers. Back in high school I had an amazing engineering teacher and when I recently went to visit him on his desk he had a dynamic speaker made out of a paper bowl, some wire, magnets and k'nex. This speaker was the most simple example of a dynamic driver I've seen and if I can get back to his office before I need to go back to school I'll replace this picture with the one I'd really like to put here.
Balanced Armature Drivers -
These drivers are much smaller then dynamic drivers can be. Although these drivers are unstable, they provide more sound using the less power then any other type of driver can. These drivers have a moving magnetic armature that is position in a way that allows them to move in the magnetic field of the permanent magnet that surrounds them while being perfectly balanced when not moving, hence the name.
When a current is run though these drivers it causes the armature to twist one way or the other, tugging or pulling on the diaphragm, this diaphragm manipulation creates the sound.