Well, for those of you who don't want to read the big paragraph I just wrote, here are the specs.
For the low voltage side, I used a modified computer power supply to deliver 12VDC to my 555 timer flyback driver (http://tacashi.tripod.com/elctrncs/555sstc/555sstc.htm), because I used a modern flyback from my old television set, it has a built in diode and produces pulsed DC. This is better for simple Tesla Coils because it means that it is easier to find capacitors that can stand up to the voltage.
The primary coil is four pieces of wood that are aligned to form a cross with space for the secondary to fit inside. The pieces of wood have holes drilled into them, and 14 AWG wire is passed through them to form the coil. The wire is stranded (not good), and was rescued for the power cord of an old microwave oven. My primary coil is very bad. Do not try to duplicate it. It needs more turns (I ran out of wire, it was too short), and to be done with more care.
The secondary is about 20m. of 30 AWG wire wound onto some 2" PVC pipe. Wind as many turns as you can!
I used an MMC (Multi-Mini Capacitor) for my tank capacitor in the circuit. An MMC is basically a large array of smaller capacitors (called snubber capacitors), that are combined to form a larger value, and higher voltage rating. I used a lot of 1.5kV, 0.02uf ceramic disk type capacitors in various combinations to produce an overall rating of about 25kV at 0.15uf. The voltage rating should be about 5kV higher then the output of your power supply (If its DC) for safety reasons. If you have an AC supply, make the voltage rating of your capacitor about 3-4 times as large as the RMS (The output voltage) voltage of your HV power supply
The spark gap was just two screws in a small piece of wood with a flexible piece of aluminum attached to one of them to regulate the distance. The gap did not get hot enough for it to be a worry. I don't think that with such a small coil you need tungsten electrodes, a rotary gap, of even a cooling fan.
I will have better images coming soon, and I hope to build a new primary/secondary coil too. This is a surprisingly easy project, that yields interesting results. The arcs were not too impressive, but I could light neon bulbs from a distance, and I will even have a picture of a foot long xenon tube I constructed being excited by this device.
The second picture is a Kirlian photograph taken with a digital camera and a Tesla Coil in my basement. I have many more, and an instructable is coming soon.
Today, I wound a new primary coil using some reclaimed wire. It was primitive (I wound it by gluing it to the old wooden primary coil), but it worked! The performance yesterday was laughable. It was barely a Tesla Coil. Now, I can adjust the primary winds using alligator clip wire and it produces centimeter long coronal discharges into the air and can light up a xenon tube that is over a foot long. I can get an inch long spark to the tip of a screwdriver, and it lights up a lightbulb with large streamers even during the day when the sun is shining directly onto it. For a twenty dollar coil that was built in a few hours, I am impressed.
-Foot long xenon tube chain
-Coronal discharge off of the end of the foot long tube (low res.)
-Detail of new primary coil
UPDATE 2: 4/5/2009
Actually, within a few days of building this coil, I wound a new secondary coil. When I tuned everything, the coil performed very nicely. It produced 1 inch long discharges into the air and 1.5 - 2 inch discharges to a screwdriver. However, I was too lazy to post anything, so this slideshow remained unchanged. I later disassembled the coil and let the parts sit around for a few months, until now. A few days ago, I reassembled it with a new primary coil, tank capacitor, and spark gap. The results were great, so I am posting some new pictures.
-A picture of the new coil
-A picture of some sparks
-A picture of the coil lighting a fluorescent tube
-A picture of my new SSTC - Made almost entirely out of free samples and scrap parts. I have since improved it and it works very well in my opinion, 1 cm sparks into the air.
UPDATE 3: 10/26/2009
I haven't been updating this much, have I? Anyhow, much has happened since the pictures you see here. I actually managed to improve this coil SIGNIFICANTLY to produce loud, bright, 6 inch long sparks from a new secondary coil. I upped the voltage of the flyback driver, and installed new capactiors and new coils. Unfortunately, the coil had big problems with arcing between the primary and secondary, and I think I could have done much better than 6" if I had been able to fix them. However, I eventually pushed it too hard, and then whole thing died. In all, this was a great experience, and I gained much knowledge and experience from it.
In other news, I also improved the Vacuum Tube Tesla Coil (VTTC) further until it gave roaring 6" long streamers. Similarly, I ended up pushing some of its components too hard, until a few rather trivial small components broke. I didn't have the resources to fix the problem then (mainly because of negative support from my parents), and so I left the project to rest. Now, only a few days ago, I ordered the replacement parts for the coil, as well as parts for a half bridge audio modulated Solid State Tesla Coil, needless to say, it should be amazing when its built (this might take a couple of weeks, but be on the lookout for more pcitures!).
UPDATE 4: 11/6/2009
I just added a picture of first light for my Solid State Tesla Coil at 24 volts (about 1/7 full power), it also plays music via PWM, so I'm going to upload a video soon too!
UPDATE 5: 1/9/10
My Mini Solid State Tesla Coil finally works properly, here is it running at 90 VAC RMS (3/4 of full power). The sparks are 7" - 8" long! Audio modulation is very impressive - more videos soon.
Here is an early audio modulation video: