I'm not going to provide a thorough explanation because several other people have already done so (refer to links below). Also, a deep understanding of Tesla coil operation is unnecessary for people wishing to build a Tesla coil. However, I will offer a short description of Tesla coil operation that should help you design and build your Tesla coil.

A Tesla coil is a resonate transformer containing a primary and secondary LC circuit. The two LC circuits are loosely coupled together. Power is supplied to the primary circuit through a transformer, which charges a capacitor. Eventually the voltage across the capacitor will increase sufficiently to short a spark gap. The capacitor will discharge through the spark gap and into the primary coil. The energy will oscillate back and forth between the primary capacitor and primary coil inductor at high frequencies (typically 100 - 300 kHz). The primary coil is coupled to an inductor in the secondary circuit, called the secondary coil. Attached to the top of the secondary coil is a top load that provides capacitance for the secondary LC circuit. As the primary circuit oscillates, power is induced in the secondary coil where the voltage is multiplied many times. A high voltage, low current field develops around the top load and arcs of lightning discharge in a sweet display of awesomeness. The primary and secondary LC circuits must oscillate at the same frequency to achieve maximum power transfer. The circuits in the coil are usually "tuned" to the same frequency by adjusting the inductance of the primary coil.

For a more detailed description I recommend the following resources:

The Tesla Coil by Chris Gerekos
Chris has written an outstanding paper offering a very detailed and technical explanation of Tesla coil operation. He also shares his experience constructing the "Zeus" Tesla coil. The paper is in pdf format.

Tesla Coil Theory by Terry Blake http://www.tb3.com/tesla/theory.html

Richard Burnett's page http://www.richieburnett.co.uk/tesla.shtml

The power supply is a high voltage transformer used to charge the primary capacitor. Neon Sign Transformers (NSTs) are the most common power supply used in small to medium sized Tesla coils. For the rest of the guide I'll refer to the power supply transformer as a NST.

These calculations will be used to determine the optimum sized primary capacitor (in the next section).

NST VA = NST V_out × NST I_out

NST Impedance (ohms) = NST V_out ∕ NST I_out

We aren't required to calculate the NST watts, but it may be helpful for selecting resistors, fuses, wire gauges, etc.

NST Watts = ((0.6 ∕ NST VA ^0.5) + 1) × NST VA

A Power Factor Correction (PFC) capacitor can be wired across the NST input terminals to correct the AC power phase and increase efficiency. The optimum PFC capacitance is found with the following equation.

PFC Capacitance (uF) = NST VA ∕ (2 × pi × NST F_in × (NST V_in²)) × 1,000,000

Where:
F_in is input frequency
pi = 3.14

The primary capacitor is used with the primary coil to create the primary LC circuit. A resonate sized capacitor can damage a NST, therefore a Larger Than Resonate (LTR) sized capacitor is strongly recommended. A LTR capacitor will also deliver the most power through the Tesla coil. Different primary gaps (static vs. sync rotary) will require different sized primary capacitors.

Primary Resonate Capacitance (nF) = 1 ∕ (2 × pi × NST Impedance × NST F_in) × 1,000,000,000

Primary LTR Static Capacitance (nF) = Primary Resonate Capacitance × 1.618

Primary LTR Sync Capacitance (nF) = 0.83 × (NST I_out ∕ (2 × NST F_in) ∕ NST V_out) × 1,000,000,000

The secondary coil is used with the top load to create the secondary LC circuit.

The secondary coil should generally have about 800 to 1200 turns. Some secondary coils can have 2000 turns. Magnet wire is used to wind the coil. There's always a little space between turns, so the equation assumes the coil turns are 97% perfect.

Secondary Coil Turns = (1 ∕ (Magnet Wire Diameter + 0.000001)) × Secondary Wire Winding Height × 0.97

The capacitance of the secondary coil will be used to calculate the secondary LC circuit resonate frequency. Coil dimensions are given in inches.

Secondary Capacitance (pf) = (0.29 × Secondary Wire Winding Height + (0.41 × (Secondary Form Diameter ∕ 2)) + (1.94 × sqrt(((Secondary Form Diameter ∕ 2) ³) ∕ Secondary Wire Winding Height))

The height to width ratio should be about 5:1 for small Tesla coils, 4:1 for average sized Tesla coils and about 3:1 for large Tesla coils. Refer to the secondary coil construction section to define small, average and large.

Secondary Height Width Ratio = Secondary Wire Winding Height ∕ Secondary Form Diameter

The length of the secondary coil is used to calculate the wire weight. In the past it was thought that the secondary coil wire length should match the quarter wave length of the Tesla coil's resonate frequency. However, it has since been determined that it's unnecessary.

Secondary Coil Wire Length (ft) = (Secondary Coil Turns × (Secondary Form Diameter × pi)) ∕ 12

Magnet wire is typically sold by weight, so it's important to know the required wire weight.

Secondary Coil Wire Weight (lbs) = pi × ((Secondary Bare Wire Diameter ∕ 2) ²) × Secondary Coil Wire Length × 3.86

The inductance of the secondary coil will be used to calculate the secondary LC circuit resonate frequency.

Secondary Inductance (mH) = ((((Secondary Coil Turns ²) × ((Secondary Form Diameter ∕ 2) ²)) ∕ ((9 × (Secondary Form Diameter ∕ 2)) + (10 × Secondary Wire Winding Height)))) × 0.001

The top load is used with the secondary coil to create the secondary LC circuit. Generally a toroid or sphere shape is used. The ring diameter refers to the ring in a toroid shape. The overall diameter refers to the widest length from edge to edge of a toroid shape. I've found several equations for different sized top loads. Without knowing which is the most accurate in any case, I use the average of all the equations.

For large or small toroids with ring diameter < 3" or ring diameter > 20", use the average of three toroid capacitance calculations.

Toroid Capacitance 1 (pf) = ((1 + (0.2781 − Ring Diameter ∕ (Overall Diameter))) × 2.8 × sqrt((pi × (Overall Diameter × Ring Diameter)) ∕ 4))

Toroid Capacitance 2 (pf) = (1.28 − Ring Diameter ∕ Overall Diameter) × sqrt(2 × pi × Ring Diameter × (Overall Diameter − Ring Diameter))

Toroid Capacitance 3 (pf) = 4.43927641749 × ((0.5 × (Ring Diameter × (Overall Diameter − Ring Diameter))) ^0.5)

Toroid Capacitance (pf) = (Toroid Capacitance 1 + Toroid Capacitance 2 + Toroid Capacitance 3) ∕ 3

Ring diameter between 3" and 6"

Toroid Capacitance Lower = 1.6079 × Overall Diameter ^0.8419

Toroid Capacitance Upper = 2.0233 × Overall Diameter ^0.8085

Toroid Capacitance (pf) = (((Ring Diameter − 3) ∕ 3) × (Toroid Capacitance Upper − Toroid Capacitance Lower)) + Toroid Capacitance Lower

Ring diameter between 6" and 12"

Toroid Capacitance Lower = 2.0233 × Overall Diameter ^0.8085

Toroid Capacitance Upper = 2.0586 × Overall Diameter ^0.8365

Toroid Capacitance (pf) = (((Ring Diameter − 6) ∕ 6) × (Toroid Capacitance Upper − Toroid Capacitance Lower)) + Toroid Capacitance Lower

Ring diameter between 12" and 20"

Toroid Capacitance Lower = 2.0586 × Overall Diameter ^0.8365

Toroid Capacitance Upper = 2.2628 × Overall Diameter ^0.8339

Toroid Capacitance (pf) = (((Ring Diameter − 12) ∕ 12) × (Toroid Capacitance Upper − Toroid Capacitance Lower)) + Toroid Capacitance Lower

Small Tesla coils may use a sphere shaped top load.

Sphere Capacitance (pf) = 2.83915 × (Sphere Diameter ∕ 2)

Sphere Diameter = Circumference ∕ pi

The total secondary capacitance includes the capacitance in the secondary coil and the capacitance of the top load. If you use multiple top loads, add their capacitance to calculate the total secondary capacitance. The total secondary capacitance will be used to calculate the secondary resonate frequency.

Total Secondary Capacitance = Secondary Coil Capacitance + Top Load Capacitance

The secondary LC circuit resonate frequency will be used to calculate the amount of primary coil inductance required to tune the Tesla coil.

Secondary Resonate Frequency (kHz) = 1 ∕ (2 × pi × sqrt((Secondary Inductance × 0.001) × (Total Secondary Capacitance × 0.000001)))

The primary coil is used with the primary capacitor to create the primary LC circuit. The primary coil is also responsible for transferring power to the secondary coil.

First, we should determine the inductance required to tune the Tesla coil. After the inductance is calculated for each turn on the primary coil, we can use the "Needed Primary Inductance" value to indicate the proper turn where we should tap the primary coil. It will also indicate the minimum number of turns required in the primary coil. Of course, the primary coil should have several extra turns - just in case you need them.

Needed Primary Inductance (H) = 1 ∕ (4 × pi² × (Secondary F_res × 1000)² × Primary Capacitance × 0.000000001)

Where:
F_res is the Secondary Resonate Frequency

The following equations will calculate the dimensions of the primary coil and the inductance of the coil at each turn. Unfortunately, you may need to run through these equations several times to determine the inductance at each turn. Of course, the TeslaMap program can quickly and easily calculate the dimensions and inductance of the coil out to 100 turns.

Primary Coil Hypotenuse = (Primary Coil Wire Diameter + Primary Coil Wire Spacing) × Turns

Primary Coil Adjacent Side = Primary Coil Hypotenuse × cos(toRadians(Primary Coil Incline Angle))

Primary Coil Diameter = (Primary Coil Adjacent Side × 2) + Primary Coil Center Hole Diameter

Primary Coil Height = Primary Coil Wire Diameter + Primary Coil Adjacent Side × tan(toRadians(Primary Coil Incline Angle))

Primary Coil Wire Length (ft) = (Primary Coil Diameter × pi) ∕ 12

Primary Coil Average Winding Radius = (Primary Coil Center Hole Diameter ∕ 2) + (Primary Coil Hypotenuse ∕ 2)

Primary Coil Inductance Flat = (Primary Coil Average Winding Radius ² × Turns ²) ∕ ((8 × Primary Coil Average Winding Radius) + (11 × Primary Coil Hypotenuse))

Primary Coil Winding Radius = (Primary Coil Center Hole Diameter ∕ 2) + (Primary Coil Wire Diameter ∕ 2)

Primary Coil Inductance Helix = ((Turns × Primary Coil Winding Radius) ²) ∕ ((9 × Primary Coil Winding Radius) + (10 × Primary Coil Height))

The inductance of a conical shaped coil is found by calculating the inductance of a flat and helical coil and using the average of the two coils weighted by the incline angle.

Angle Percent = 0.01 × (Primary Coil Incline Angle × (100 / 90)

Angle Percent Inverted = (100 − (Angle Percent × 100)) × 0.01

Primary Coil Inductance (uH) = (Primary Coil Inductance Helix × Angle Percent) + (Primary Coil Inductance Flat × Angle Percent Inverted)

This is a fairly typical Tesla coil design using a static spark gap, which should be a good starting point for a small to average sized Tesla coil. This design should produce over 2 foot arcs with the specified input power.

Design Parameter	Value (standard)	Value (metric)
NST Input Voltage	120 V	240 V
NST Input Frequency	60 Hz	50 Hz
NST Output Voltage	15 kV	15 kV
NST Output Current	30 mA	30 mA
NST Watts	463 W	463 W
Primary Capacitance (MMC)	9 nF	9 nF
Primary Coil Wire Diameter	0.25 in (tubing)	6 mm (tubing)
Primary Coil Wire Spacing	0.25 in	6 mm
Primary Coil Center Hole Diameter	6 in	15 cm
Primary Coil Incline Angle	0 degrees (flat)	0 degrees (flat)
Secondary Coil Magnet Wire Gauge	24 AWG	0.5 mm
Secondary Wire Weight	1.37 lbs	612 g
Secondary Coil Winding Height	22 in	56 cm
Secondary Coil Form Diameter	4.4 in	11 cm
Secondary Coil Turns	972	972
Secondary Coil Height To Width Ratio	5:1	5:1
Toroid Ring Diameter	4 in	11 cm
Toroid Total Diameter	16 in	40 cm

With the power supply listed above (15kV) and using a static spark gap, the primary capacitance (MMC) should be about 8.6nF (calculated with the TeslaMap program). The MMC should have enough capacitors in series for a minimum voltage rating of 15kV RMS * 1.414 = 21kV peak. It's a good idea to double the peak voltage rating to about 40kV. Using 0.15uF, 2kV caps, (Cornell Dubilier 942C20P15K-F) a string of 20 wired in series would have 7.5nF at 40kV (also calculated with the TeslaMap program), which is close enough for our needs.

This design was generated with the TeslaMap program. The design file is available for download. Once downloaded, you can open, edit and save the design with the TeslaMap program. A complete design summary has also been exported to a txt file from the TeslaMap program, which can be downloaded and viewed in any text editor and most browsers.