Coil noise is, as its name suggests, caused by electromagnetic coils. These coils, which may act as inductors or transformers, have a certain resonant frequency when coupled with the rest of the electric circuit, as well as a resonance at which it will tend to physically vibrate.
As the wire that makes up the coil passes a variable current, a small amount of electrical oscillation occurs, creating a small magnetic field. Normally this magnetic field simply works to establish the inductance of the coil. However, this magnetic field can also cause the coil itself to physically vibrate. As the coil vibrates physically, it moves through a variable magnetic field, and feeds its resonance back into the system. This can produce signal interference in the circuit and an audible hum as the coil vibrates.