The tank circuit is the part that is in between the transformers, BUT it also includes the first part of the air core transformer, commonly called the primary coil. This is one of the most important areas of a tesla coil. It is here that the low frequency AC is turned into high frequency AC. There is enough capacitance and inductance to drive the circuit into an severly underdamped resonse. This means that for every cycle in, you get hundreds out. It also increases the voltage slightly. You'll notice that there is a spark gap in parallel with the transformer. This is simply an air gap between two electrodes. When the voltage in the capacitor reaches a certain voltage, a spark arcs across the gap and allows the capacitor to discharge. This happens hundreds of times a second and produces intense voltage surges.
This is where all of the excitement happens. As the voltage pulses through the tank circuit, it is picked up by the other side of the air core transformer, called the secondary. This steps the voltage up more than any other part of the circuit. The top of the transformer is the area of highest potential, thus where the lightning comes off. A torus (as shown above) or a ball is commonly placed on top.
In addition to high voltage, there are also some other interesting effects produced by the coil. Intense magnetic and electrical fields suround the coil. Radio waves are also emitted. In conjuction with each other, these field can do erie things like light up floresent bulbs (even if they are burnt out) and disrupt nearby communications.
