First things first:

DISCLAIMER: I am not responsible for any injuries or property damage that may befall you from following this instructable. High voltage electricity can be DANGEROUS and should only be worked with at your own risk. Proper safety precautions should always be followed.

That out of the way, welcome to my first instructable. Seeing as this is my first, any suggestions for improvements are greatly appreciated. Just go easy on me.
This is intended to be a how-to guide for a newbie to high voltage (like myself) looking for a quick, cheap, and relatively safe project. Although this is not a true tesla coil, as it does not utilize a resonant air-core transformer or operate at high frequencies, in effect it is similar. It still throws out plasma discharges from the top load and about 3.5 centimeter arcs to ground. Estimated output is about 100kv.

Step 1: Parts and Pieces

There aren't many parts to this build, and most can easily by scrounged from old TVs and other electronics or be bought for cheap. The following is needed:

Bug zapper racquet: This can be purchased from Ocean State Job Lot for about 5 bucks, and is nifty for fending of mosquitos or high voltage experiments. There are probably other types of devices very similar, but I would recommend finding the racquet pictured to insure the internal circuitry is the same.
Flyback transformer: Any flyback transformer will do, though the bigger the better. Don't kill yourself looking for an old non-rectified design, since there are no benefits of it for this circuit.
Random assorted hardware: This circuit requires a spark gap to be constructed. The design of the spark gap can vary, as long as the two ends where the arc jumps is rounded, and the gap adjustable. For mine, two Erector set brackets were used. One had a ball bearing soldered to it, the other a nut over top the hole, so a bolt with an acorn nut on the end can be threaded through. See the attached picture for the details.
2xAA battery holder: Can be purchased from Radioshack or the bug zapper handle can be used to hold the batteries.

Optional:
Additional Capacitors: Should be rated for at least 1.6 KV. The Bug zapper already contains one, but for bigger sparks more can be used.
Toggle Switch: The switch on the board of the bug zapper can be difficult to use, and because of the design of the bug zapper circuit, floats at high voltage, leading to a shock hazard when it is exposed. Because of this, a new switch is recommended.
Pen body or other plastic tube: To elevate the top load
Top load: I used a ball bearing, but anything smooth and without sharp edges or points can be substituted.

Of course, solder and a soldering iron as well as other general tools are needed, and wire for connecting everything together

Step 2: Dismantle the Zapper

The bug zapper is easy enough to open. First, pry off the battery cover, and then remove the screws. There are two up near the head of the racquet, two near the bottom of the battery compartment, and another at the top of the battery compartment. Once removed, the back half of the handle can be lifted off, exposing the back of the circuit board. Remove the screw in the middle of the circuit board, and snip the wires running to the head of the racquet as close to the head of the racquet as possible, and snip the wires where they attach to the battery contacts. Now that the board is removed, the rest of the racquet is not needed.

Step 3: Prepare the Zapper Circuit

Now that the circuit is removed, it has to be slightly modified for our needs. First, remove the original momentary push switch, and in its place solder a jumper. Next, remove the negative battery wire from the board, and solder in its place the lead from the AA battery holder. Solder the positive battery wire on the board to the normally open lead on the toggle switch, and the positive lead from the battery holder to the common lead on the toggle switch. If you have extra capacitors, these can be used to make a capacitor bank. if you go this route, desolder the capacitor from the board, and set aside with the other capacitors. If you choose to not do this, leave the capacitor in its place. One of the black wires from the board's output can also be removed, since it is not needed. If you choose to make a capacitor bank, see below. Otherwise, the board is all set. The final product with capacitor bank is showed below, mounted on a piece of painted mdf.

Capacitor Bank:
This is relatively simple to make. Find as many high voltage capacitors as you want to use and wire them in parallel. In my case, I chose to use six, for no apparent reason. They can be mounted on a perf board as shown for a neater appearance.

Step 4: Spark Gap

One of the wires from the zapper capcitor/capacitor bank feeds directly to the flyback transformer, which will be addressed in the next step, the other through the spark gap. The spark gap works to allow the capacitors to charge to the point when the electricity jumps the gap, and continues into the flyback. This creates short, powerful pulses to feed the flyback. The design of the spark gap can vary, but there are some general requirements:
It has to be adjustable for the width of the gap.
The ends of each electrode should be rounded.
The rounded shape is to prevent corona leakage between the electrodes. For my spark gap, one electrode is a ball bearing, the other a bolt with an acorn nut on the end. The electrodes are then mounted on Erector set pieces, and each nailed to the mdf that my whole setup is mounted on. the bolt can be screwed in or out to adjust the width of the gap. The wider the gap, the slower but more powerful the pulses, the narrower the gap, the faster but weaker the pulses. The gap then feeds into the flyback transformer.

Step 5: Flyback Transformer

This is easily the most time consuming and tedious part of the build. While many other people might rewind their own primary coils on the flyback, I prefer to use the ones already available, since they are already nicely potted in the flyback. Unlike most flyback driver circuits, which use a primary and feedback coil, this just uses one primary coil. To find the primary coil, its down to trial and error. Using a multimeter, measure the resistance across each pair of pins. I find that in a majority of flybacks, the primary coils (as there are usually more than one) are situated so that their inputs are next to each other. That being said, this is not always the case. As you measure across each set of pins, take note of their resistances, as the one with the lowest resistance has the fewer number of coils. This is the one we are after. However, make sure that this is an independent coil, and that there isn't a third pin connected to it. Once this coil is located, the secondary coil needs to be located. Part of this is already done, since one "pin" is the fat (usually red) wire that comes out of the top of the transformer and has a suction cup on the end. The method for locating the second pin is relatively crude. Connect a 9 volt battery to one of the primary coil pins with an alligator clip, and to the other primary coil pin, connect an alligator clip. Don't connect this alligator clip to the battery yet. Take fat the red wire, and with the suction cup removed and the end stripped, place it close to one of the unused pins. Tap the disconnected terminal of the battery with the loose alligator clip, and look for a spark between the wire and pin. If there is none, move it closer and try again. If there is still no spark, move onto the next pin. If the wire doesn't spark to any of the pins, reverse the polarity of the battery and try the whole process again. Eventually, you will come across the pin you are looking for. Before disconnecting the battery from the flyback, take note of the polarity of the primary coil pins. If you are using one of the new flybacks, the polarity is important, since they contain a rectifier and voltage multiplier circuit. Once the primary coil is located, solder two long wires to it, and to the pin that the fat red wire sparks to, solder another wire. Then, just to be safe, pot the pins in hot glue. Make sure to use plenty of glue, and fill all gaps and spaces. This prevents unwanted arcing. Once this is done, the coil is all set.

Step 6: Putting It All Together

The flyback is now ready to be wired into the rest of the circuit that we prepared. In my circuit, the positive output from the capacitor bank goes thought the spark gap, then to the flyback. The negative output from the capacitor bank goes directly to the flyback. In this way, the spark gap is wired in series with the flyback. Once this is all set, the circuit is ready to be tested. flip the circuit on, and the red LED that was already on the zapper board should light up. This means the circuit is running and the capacitors are being charged. If you don't get a spark across the spark gap, check the width of the gap. If the electrodes are touching, back the bolt out (or however your spark gap is set up) until a spark is achieved, or if they are too far apart, make the gap smaller. DON'T MAKE ADJUSTMENTS TO THE SPARK GAP WHILE THE CIRCUIT IS ON!!!!! If you do so, you will be shocked. Once you have a spark, put the fat red wire close to the wire soldered to the other pin of the secondary. You should have an arc jump the gap. If not move the wires closer. If you get an arc from the secondary coil of the flyback, give yourself a pat on the back, your circuit is done! if not, time for troubleshooting. Check all connections, make sure the capacitor bank is charging by using a high voltage multimeter to check the voltage across the capacitors, check the spark gap width, check the polarity of the primary coil connections to the flyback, and check to make sure you are using the proper pins. Once the circuit is working, it's time to package it all up. Below, the picture is after its been mounted to a piece of mdf and a top load added, which is addressed in the next step.

Step 7: Top Load and Mounting the Circuit

To make the circuit function more like a classic Tesla Coil, one end of the secondary coil needs to be grounded. This is done by simply attaching it to a grounding post or cold water pipe. The pin of the secondary that should be grounded is the one on the bottom of the flyback. The fat red wire is connected to the top load. The top load is simply something metal and smooth, without any edges or points. A large ball bearing works well for this. I then glued the top load to a plastic pen body, with the wire running up the inside. to make the whole set up neater, it was mounted on painted mdf (medium density fiberboard). The wire running to the top load enter the side of the board, then takes a right turn up inside the pen body. The other wire from the secondary coil enters the board from the side, but then connects to a binding post, so that a grounded wire can be attached to it. You can choose to do it as I have done, or mounted everything in a box, or however.

Using the "Tesla Coil": Turn the switch on, and with a grounded wire attached, plasma discharges will leave the top load. Because the discharges are low amperage, they are difficult to see in the light. In a dark room, once your eyes adjust, they are visible as white, mini lightning bolts. A wire can be attached to the grounding post, and placed near the top load, so it arcs to it. The longest arc I have recorded was about 3.6 cm long. The spark gap can be adjusted to achieve different results as well. Making the gap bigger leads to fewer pulses, as low as 1 a second, but leads to the most powerful discharges out of the top load. This is best when trying to achieve the largest arc to ground. Making the gap smaller leads to faster, but less powerful pulses. This is best for making plasma discharges into the air. Making the gap too small, however, will severely weaken the discharges. Finally, don't run the circuit for too long, no more than about 30 to 35 seconds at a time, as this can lead to the zapper circuit overheating and failing.

Congratulations on your new AA battery powered "Tesla Coil"!. Have fun with it, show it off to family members and friends, and experiment with high voltage! Remember, just remember to use common sense and be safe.