Here is a web site you should visit for more information on DIY (Do It Yourself) Tesla coil construction. There is also a
Tesla Coil Web Ring that links such sites together.
The components in your video are just
part of what you need to construct a working Tesla coil, but they are the most important and frequently the most difficult to acquire parts. The neon sign transformer with its center-tapped, high-voltage, current-limiting secondary winding, in conjunction with a capacitor bank and arc gaps, is perhaps the most important component. It serves to charge the capacitors until the arc-gap "fires" and discharges them though the arc gaps into the primary coil.
During the discharge through the arc, high-frequency oscillations are created in the primary coil that are coupled inductively into the vertical coil, which is resonated also to these frequencies with a large toroid capacitance hat. The toroid also serves the function of "smoothing" the high-tension electrical field, thus enabling larger voltages to occur before the air around the toroid breaks down and "streamers" begin to form.
Many "modern" Tesla coils do not employ arcs to create the current in the primary winding, which is what the copper tubing is for. However, some "purists" will insist that a real arc, one that hisses loudly, is necessary for a "real" Tesla coil. Much has been written about how to build real arcs, including rotary (motor-driven) multiple arc electrodes. The primary of a Tesla coil is often "flat wound" in a spiral and placed, centered, near the bottom of the secondary. Tesla coils that do not employ arcs use other means to discharge the capacitor or capacitors that the high-voltage neon sign transformer charges. This can be problematical for the DIYer because arc current often exceeds several hundred amperes, while the arc gap must hold off several thousand volts before arcing. It is difficult to find either vacuum tubes or solid-state components that will operate under such conditions at affordable prices. However, if you are wllling to scale down your coil, at least initially, vacuum tubes that were formerly used in the fly-back circuits of color televisions can serve nicely to drive the primary winding.
It helps to have a strong mathematical background and a thorough grounding in electrical and electronics theory before embarking on Tesla coil adventures. Keep careful notes describing your successes and failures so you will (eventually) know what works and what is a waste of time and money. Of course, the layman will say any Tesla coil is a waste of time and money, because all it does is create arcs and sprarks. Some newbies will buy into the belief that "Tesla knew something" and all they have to do is re-discover it to become famous, feeelthy rich, or whatever their favorite fantasy happens to be. But if all you want to do is build Tesla coils because it is a fun thing to do... well, that's reason enough to do so. Electronics (and electricity) is supposed to be FUN!
Edit: BTW, the high-frequency currents created in the primary winding MUST be prevented from finding their way back into the mains wiring through the neon sign transformer. That is what the little silver box identified as a power line filter is for. It appears to be woefully too small to be effective for as "serious" a coil, as implied by your selection of primary coil and secondary coil components, as you are attempting to build.
