A Tesla coil uses a spark gap to drive oscillations in a resonant transformer circuit to produce high voltages at low currents. It works by using a high voltage supply transformer to charge a capacitor, which then discharges through the primary coil via a spark gap, generating radio frequency oscillations. The oscillating magnetic field induces currents in the secondary coil, causing voltage buildup in the coil and attached electrode. Tuning the primary and secondary circuits to resonance maximizes energy transfer and output voltage.