Resonance is the event that two systems are moving at the same frequency (Resonant frequency)

If the man pushes the swing at the right instance relative to the swing's motion, the swing will move farther. These two systems, the man pushing and the swing moving, are resonating.

A change in magnetic field produces a change in current, following the left-hand rule:

A change in current produces a change in magnetic field, following the right-hand rule. In practical aplication, a coiled wire will add up the magnetic fields from each turn like so: 

Combining Faraday's and Ampere's law, we can use changing currents on the primary coil (left), to create a changing magntic field. Now with the secondary coil (right) coming in contact with that changing magnetic field of the primary coil, we can wirelessly form a changing current on the secondary coil.

Depending on the ratio between the coils, transformers can step-up or step-down voltage, inversely to current. 

The power supply takes in power from the outlet and steps it up to charge the capacitor in the primary circuit .

This power supply must step up the 120Vac 60Hz from the wall, to a higher AC voltage for the capacitor to quickly build charge.

It's common for these power supplies to reach around 6-10kVac as that is the highest voltage our wallets can typically survive. This is why microwave ovens are highly valuable amongst high voltage hobbyists. 

The power supply(60Hz) charges the capacitor. Once the capacitor is charged, there's enough voltage for the spark gap to conduct (Paschen's Law). 

When the spark gap conducts, the capacitor and the inductor resonate back and forth at super high frequencies(~100kHz!), charging and discharging each other, until the losses are great enough to stop conducting through spark gap(~300Hz).

Note the primary circuit has 3 different frequencies at play.

•  60Hz from the power supply

• ~300Hz as the spark gap fires

• ~100kHz as the LC circuit oscilates

Using the primary and secondary side coils as a transformer, the energy from the primary circuit is transfered to the secondary, increasing voltage at the cost of current. For optimal power transfer, it's desired to have the energy of primary circuit, equal to te energy of secondary. By doing so, the peak current in the secondary coil(inductor) will occur right at the peak change in magnetic field of the primary coil, thus maximizing the energy transfer. 

Like the man pushing the swing at the right time to maximize the kid's momentum, the oscilations between charging and discharging of the primary and secondary coils must act the same.

This means the primary and secondary circuits must match frequencies, to push the most change-in-magnetic field, into electric current (and thus voltage)

This is where the lightning happens, and it's essentially a scaled up version of the primary circuit in reverse.
In the primary, a spark-gap allows a capacitor to discharge through an inductor in resonance.
In the secondary, a huge inductor charges a huge capacitor, allowing a huge spark-gap to occur between the capacitor's positive and negative terminals (top load and ground)