Tesla Coils were the invention of Nikola Tesla
in the mid to late 1800's. They do NOT produce static electricity
at all but are really air-core resonating transformer
They normally consist of what is commonly called a tank circuit. This is an Alternating Current input which charges a capacitor. When the voltage hits its peak a spark gap closes the circuit, causing the energy in the capacitor to be dumped into a coil known as the primary coil.
This causes the current to "ring" through the coil back and forth. This circuit is tuned such that its frequency is close to the resonant frequency of the secondary circuit.
This causes current to be induced into a large coil known as the secondary coil. The circuit consists of the coil, which is grounded at one end, and discharge capacitor at the end, which is normally a torroid. This capacitor has only 1 plate, the other side of the capacitor being the ground itself!
The end result is huge sparks flying through the air and the ability to light up flourecent tubes steadily at a distance of several feet.
There are two really main varients of this coil. Namely the ubiquitous 1/4th wave (which is actually what I have just described) and the 1/2 wave.
You can think of the coil as a string on a musical instument. If you pluck the string then it will vibrate with certain frequency, which is based on the length of the string. This is a "resonant frequency".
In truth there are infinite numbers of these, and all are multiples of the main one. The rining of the capacitor and inductor is similar in this regard. The 1/4th wave coil sets up a resonance of 1/4 of a wave... 1 wave in alternating current is where it goes from zero, to full positive, back to zero, and then to full negative and back to zero. We set up a frequency that does 1/4th of that.
So what we end up with a node at the grounded end, and the peak voltage at the other end of the coil. Each time the spark gap triggers (which happens several times a second) it is like "striking the string" to put more energy into it, thus we get a cumulative effect (up to the limits imposed by inefficiency in the system and discharges) this is known as ferranti rise.
Now a half wave coil is different, because we end up with a node in the middle.
We end up with a zero node in the middle, and opposite maximums of the wave form at each end of the coil, which gives a huge potential difference!
These coils are usually set to terminals that can arc to each other procucing very powerfull arcs between them!