A Tesla Coil is a specially designed transformer, technically termed an "air-core resonant transformer". It consists of a primary winding, with relatively few turns, and a secondary winding with hundreds, or even thousands, of turns. As with all transformers, the primary and secondary windings are physically arranged so that electrical energy may be transferred between them by transformer action - a changing current flowing in one winding induces a changing voltage in the other. Unlike low frequency power transformers, no ferromagnetic "core" is used, and the windings are "loosely coupled" - typically only 10-20% of the primary's total magnetic field links with the secondary winding. 

However, there are even more profound differences. The primary winding is connected to a comparatively large capacitor. The primary winding inductance (Lp) and primary tank capacitor (Cp) combine to form a tuned circuit which will want to oscillate at a specific frequency (called the natural resonant frequency) which is uniquely determined by the values of Lp and Cp. In a similar fashion, the secondary winding's inductance (Ls), combined with the self-capacitance of the secondary and the self-capacitance of the terminal connected to the top of the secondary (total capacitance Cs), form another tuned circuit. By design, the resonant frequencies of the secondary and primary circuits are adjusted ("tuned") so that they are identical. The  operating frequency of a typical Tesla Coil is typically in the range of 40 kHz (for very large systems) to well over 1 MHz. 

Nikola Tesla discovered that tuned resonant transformers exhibited a number of unique properties:

• Unlike most transformers, output voltage is virtually independent of the turns-ratio of the secondary versus the primary! Instead, it is a function of the relative ratio of inductances or capacitances between the primary and secondary. Extremely high voltages can easily be generated.

• The high operating frequency, combined with high output voltages, cause quite unique gaseous ionization, skin-effect,  and displacement current effects. 

• Tesla discovered that he could transmit high frequency energy (for radio communications or wireless energy transfer) with much greater efficiency than the comparatively primitive methods proposed by his contemporaries, such as Hertz, Lodge, and Marconi.