The idea is simple, really. Make a transformer with a huge step-up ratio, (secondary winding has far more turn than primary winding), and tune both windings to resonance at the same frequency. If, under these conditions, a pulse is applied to the primary winding (for example, by means of a spark), the voltage across the secondary will rise to level at which substantial discharges can be observed, usually in the form of light flashes, arcs and coronas.
Meanwhile, it appears that many hobbyists find pleasure in building such a Tesla transformer, and use it to create wonderful spark showers. Indeed, it has become something like a sport to ‘draw’ the largest artificial flashes of lightning. Tesla’s own record, however, has not yet been broken or even equalled — in 1899, his experimental setup in Colorado Springs produced an arc of 41 metres (approx. 125 ft), wrecking the electricity supply in the whole area.
As an indication of the popularity of the Tesla transformer, there are more than a million results on Google for the term tesla coil. From the long lists we picked some examples for those of you whose curiosity has been aroused. On Stefan’s Tesla pages we found an extensive description of the structure of a Tesla coil, complete with lots of tables, diagrams and equations. If you want something a little more practical, we’d say the products on Tesla Coil Plans, Parts, Kits page are a better starting point.
Meanwhile a couple of companies have also entered the field. The activities are mostly construction and demonstration. A typical example is Tesla Systems Research. Very spectacular!
If you want to know more about the great inventor himself, lots of relevant information may be found at Nikola Tesla Wikipedia page.