The circuit itself is fairly simple. The transformer steps up 120 volts AC to 12kV AC. This voltage is high enough to break down the air and cause an arc to form between the two electrodes where they are closest together.
The arc heats and ionizes the air, which then rises. This hot ionized air is a lower resistance path between the two electrodes, and so the arc continous to follow this path. The arc rises along with the hot ionized air, until it reaches the top of the electrodes where it cannot rise any higher. The arc then extinguishes, and forms again at the bottom where the electrodes are closest together.