How does it work: A Tesla coil is a resonate transformer consisting of a primary and secondary LC circuit. Power is supplied to the primary circuit through a transformer. A capacitor in the primary circuit will begin to charge. Eventually the voltage across the capacitor will increase sufficiently and cause a short across a spark gap. The spark gap will allow the capacitor to discharge into an inductor called the primary coil. The primary inductor 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 acts as a capacitor. As the primary circuit oscillates, power is transferred to the secondary coil where the voltage is increased and arcs of lightning are discharged form the top load. The primary and secondary circuits must resonate at the same frequency to achieve maximum power transfer from the primary to the secondary. The circuits in the coil are usually "tuned" to the same frequency by adjusting the inductance of the primary coil.
How did I make it: Firstly, I made the base out of a plastic Pepsi stand, then I started adding all the components to the base. I assembled the capacitor bank (group of connected capacitors) on a paper towel roll. I coiled some (a lot actually) of wire on a PVC pipe to make the secondary coil. I made the primary coil out of some thick copper wire, then a used some sort of plastic tubing (used for arranging wires) to hold the primary coil in place. I improvised the 'top load' or shiny metal thingy as I like to call it, out of an air-conditioning duct (a very small one).
How powerful is it: According to calculation, it outputs a whopping 615000 volts, consuming 1 amp at 220 volts.
Videos of the Tesla coil in action: