Originally from an email to a friend.
What follows is Tesla Coil construction advice from a rank amateur, with no warranty as to correctness or suitability for any purpose whatsoever - ok, my lawyer's not looking over my shoulder anymore.
First, what do you have available? Tesla coils need a hv source, capacitor, two coils and a top load.
HV source - old neon sign transformers are a good starting point. You will need one without a GFCI, as this safety feature makes them useless for a TC. Check sign repair shops for old throwaways. Another good source is the transformers out of old microwave ovens - these can be had free if you watch for them in the trash. They will likely require either a matched set (difficult to find, even in identical microwaves), or a voltage doubler circuit. I have a few, condition unknown. Automotive ignition coils are also possible, but these vary so much that it is hard to recommend.
Control circuit - consider the available power (volts and amps) at your site, as well as a good ground not connected to wall ground - consider driving at least a 4 foot long copper rod into the ground as close to the coil as possible. You may be able to substitute a drain pipe or large copper sheet. Keep electronics on different circuits, or better yet, not plugged in. Good practice puts an RF choke, lightning arrester, or similar device across the wall input to prevent TC high frequencies from backfeeding into the line. To protect your HV transformer from the higher voltages in the TC secondary, put a "Terry Filter" across the HV out. Power switching should at minimum include a deadman switch - must be held closed to provide power to the HV circuit. Don't work on them alone, or when it's wet.
Capacitor - the simplest by far is the Geek Group bucket capacitor: one clean 5 gallon bucket with lid. Fill with one dozen long neck 12 oz. bottles (IBC root beer works, don't do like I did and wander roadsides collecting beer bottles - new bottles last better, with less awkward explanations). Cook up some brine - as much ordinary salt as you can dissolve in boiling water (watch the cookwear - brine is corrosive in some pans, and plastic containers will taste of salt forever after). Fill bottles 2/3 with brine, and the same depth around the bottles. Top off bottles with mineral oil (best, motor oil will work as well), fill interstitial space to same level with oil. Make u-shaped wires (scrap 12ga copper works) to connect all the bottles. Two long wires make the two leads to the capacitor - one in the bottle network, one in the space. Put the leads through the lid at a suitable distance apart from each other, then seal lid. Store with leads shorted, as caps can spontaneously build up charge. Build a jumpstick for discharging - 5 feet of polyurethane coated pvc pipe with a conductor on one end. Cap design is 'rated' @ ~12 picofarads and 80,000 volts. Test capacitance value with multimeter or test circuit.
The geometry of the primary, secondary, and top load will depend on the components above. Primaries are typically 8-25 turns of copper tube or heavy wire - I've used 1/4" wire - in a flat or helical spiral. Secondaries are hundreds of turns of enameled magnet wire in a 3 to 5:1 aspect ratio on a sealed form (the same urethaned pvc pipe is a favorite). Dryer duct, chicken wire, any metal conductor can be formed into a top load.
To calculate the resonant design of your coils from the other components, I use this:
It's a little daunting, and can take several minutes on old browser/pc combos. Basically, you try to match the resonant frequencies of the secondary + topload/ground circuit to the primary/capacitor circuit (frequently called the tank circuit).
http://www.pupman.com/ - Home of the Tesla Coil Mailing List - search the years of archives for questions.
That's a lot to get started with - go find a bucket, 12 bottles, a microwave or neon sign transformer and go!