First the Safety.
Do not touch the electrodes of the ladder during operation and be very careful with your power supply. most NSTs can be lethal if not handled with care so take caution! Be sure to use appropriately rated wire, and to unplug your supply if you are going to touch the wires or electrodes. Do not trust power switches! Last, do not touch the electrodes shortly after use even with the power supply turned off, They get very hot and will burn you. Make sure there are no flammable items or substances near by when you use it.
You will need:
A power supply
I used a 6kv 30ma non GFI neon sign transformer for this, but you can also use a flyback transformer if you made a smaller ladder, or maybe even a microwave oven transformer. The important thing to remember if you are going to use a NST is that it must be NON GFI otherwise it will not arc across the two electrodes. Remember to use wire rated to the voltage of your power supply to hook it up.
A simple piece of wood should suffice but the best choice is probably a known insulator such as a base made from a plastic cutting board.
Any wire should work so long as it is not so thin that it will be melted by the arc. I used coat hanger wire, from the kind of hanger with the white cardboard tube on the bottom. the ends are the perfect shape for the top.
Yep, that's really all there is to it. I drilled two small holes about 1/8 inch apart in the middle of the wood for the electrodes. I then drilled two larger holes on either side of the middle for some largish screw eyes. I removed the cardboard tube from the hanger and cut two equal length pieces of wire from the hanger and removed the paint with a rasp. Finishing up I placed the two coat hanger wires in the two small middle holes at a slight angle and glued them down. I wrapped some wire around the very bottom tightly, and covered them in glue to insulate them, and attached the other end of the wires to the screw eyes. Now just attach your power supply to the two screw eyes and be amazed. As the electricity arcs across the electrodes it heats the air around it pushing the arc up until the electrodes are two far apart and the arc breaks and a new arc forms at the bottom.