Alternating current and RMS
Because generators naturally produce a sine wave, the electricity in your walls is alternating current (AC): the voltage and current oscillate smoothly, reversing many times a second. Taiwan's mains runs at about 110 V and 60 Hz. But that 110 V is not the peak of the wave — it is the root-mean-square value, the equivalent steady voltage that would deliver the same average power.
The RMS value is the peak divided by √2, so a 110 V (rms) supply actually peaks near 155 V. Average power uses the RMS values.
The point of RMS is convenience: with V_{\text{rms}} and I_{\text{rms}} you can reuse the familiar DC power formulas straight away, and P = V_{\text{rms}} I_{\text{rms}} gives the true average power delivered to a resistor. Every voltage on a mains label or a multimeter is an RMS value.
Transformers
Now the machine that made AC win. Wind two coils on a shared iron core. Feed AC into the primary and its changing current drives a changing flux around the core; that shared flux threads the secondary and, by Faraday's law, induces an EMF in it. Since both coils see the very same d\Phi_B/dt per turn, their voltages are in the ratio of their turns.
The transformer equation. Voltage scales with the turns ratio; for an ideal transformer power in equals power out, so stepping voltage up steps current down.
Energy conservation seals the deal: for an ideal transformer the power in equals the power out, so if the voltage steps up by some factor the current steps down by the same factor. Crucially, a transformer needs a changing flux, so it works only on AC — and this coupling between coils is a case of mutual inductance. That single limitation is why AC, not DC, became the language of the power grid.
The leap to electromagnetic waves
Faraday's discovery is that a changing magnetic field creates an electric field. James Clerk Maxwell added the symmetric partner — a changing electric field creates a magnetic field (through what he called the displacement current). Assembled with the earlier laws of electricity and magnetism, these become Maxwell's equations, the complete rulebook of electromagnetism.
The consequence is breathtaking. A changing electric field makes a magnetic field, whose change makes an electric field, whose change makes a magnetic field — the two fields regenerate each other and can march off through empty space with no charges, no wires, no medium at all. Maxwell found the speed of this self-sustaining wave from two constants measured in the laboratory.
The speed of an electromagnetic wave, built from the magnetic and electric constants of empty space — and it comes out equal to the speed of light.
That number came out to 3\times 10^8 m/s — the measured speed of light. The conclusion was inescapable: light is an electromagnetic wave. The very same induction that runs your toaster, pushed up to trillions of hertz, is radio, microwaves, visible light, X-rays and gamma rays — the whole electromagnetic spectrum. An electromagnetic wave is induction that has broken free of its wires.