Physicist: Stripped down to it’s most essential parts, a Tesla coil is a wire sticking out of the ground. To get sparks to fly out of the top the rest of the machine “sloshes” electrons up and down the wire.
The picture you should have in your head is a long bathtub, open to the ocean on one end. The machinery of the Tesla coil is like some dude in the bathtub sliding back and forth, splashing water (electrons) out of the closed end, while the tub is refilled from the ocean (ground).
The electricity in the primary coil is what’s doing the pushing, and the electricity in the secondary coil is what’s being pushed. To understand how the driving mechanism works requires a new metaphor and some answer gravy.
Answer gravy: To get sparks to really fly you need very high voltage (up to several million volts) at a fairly exact frequency. The current that flows up and down the secondary coil, and sloshes out the top, has a high resonant frequency (~MHz, unless the coil is ridiculously huge) that you really can’t do much about. But the current coming out of the wall has a frequency of only 60 Hz (50 Hz for our Old World readers).
So how do you change frequencies? The answer is you “pluck” the primary coil. For example: If you pick a guitar string once a second you have a frequency of 1 Hz, but the string vibrates on its own at whatever frequency it’s made for (~10 kHz).
The AC mains have a low frequency (60 Hz) while the secondary coil needs to be driven at a high frequency (~1,000,000 Hz). That means that the secondary will slosh back and forth thousands of times every time the current from the wall turns over just once. Since the fast part of the circuit is so much faster than the slow part, you can just pretend that the current from the transformer is DC (direct current = 0 Hz).
The secret to plucking is to change the circuit’s “shape” using a spark gap. Spark gaps have some pretty slick properties. They have an essentially infinite resistance until a high enough voltage is applied across them, at which point they spark (hence the name). The spark you see is the air being pulled apart and ionized. Now ionized gas is a really good conductor, so a spark is like instantly closing a switch.
Also, spark gaps are the cheapest circuit element evar. Can you cut a wire? Now you got a gap!
Also, adding spark gaps to a device is one of the quickest ways to bridge the divide between regular and mad science.
The only job that the slow part of the circuit has is to charge the capacitor (pull back the string). When the spark gap sparks (pluck!) the fast part of the circuit takes over, and the slow part is essentially ignored until all the energy is exhausted by exciting the secondary coil (string vibrates and slows).
As current flows through the primary it creates a voltage across the secondary that’s so high that electricity actually flies out of the top of the coil, despite having nowhere in particular to go. It generally takes at least several hundred thousand volts to make that happen.
The loop in the picture above forms an RLC circuit with a high resonant frequency (that matches the frequency dictated by the secondary). As the energy in this system runs out the voltage needed to maintain the spark gap (which is much less than the voltage needed to start it) is lost, and the whole thing returns to the slow, charging phase.
Since the power supply oscillates at 60 Hz, the whole system briefly turns off 120 times every second (the voltage is +, 0, -, 0, +, 0, …). For this reason Tesla coils have a very loud 120 Hz hum that sounds “staticy” and ominous, as opposed to Jacob’s ladders which are continuous, and tend to sound more like “tearing”. Connoisseurs, I’m sure, will agree.