Frequency of light versus frequency of electron vibration

Electrons and photons are quantum mechanical entities, obeying quantum mechanical equations.

What you are describing with words is a classical description of light which is controlled by classical electrodynamic equations, Maxwell equations.

An accelerating charged particle produces an electromagnetic (EM) wave. Electromagnetic waves are electric and magnetic fields traveling through empty space with the speed of light c. A charged particle oscillating about an equilibrium position is an accelerating charged particle. If its frequency of oscillation is f, then it produces an electromagnetic wave with frequency f.

Going to the quantum mechanical frame of photons , the classical light of frequency f emerges from zillions of photons of energy=h*f in a mathematically complicated way, The frequency is associated with the energy. An accelerated electron will radiate a photon according to the rules of quantum electrodynamics, and there will be a probability of generating this photon, mathematically complicated. This can be proven mathematically, but not in a handwaving way.

I'll have a go at a hand-wavy explanation which assumes that you are comfortable with the idea of an electromagnetic field. If you're not, let me know and I'll address that in an edit.

Light is ripples of electromagnetism. When people talk about the speed of light they mean how quickly the ripples travel through space. The frequency of light is the rate at which the ripples vibrate.

Most sources of light produce ripples with a mix of frequencies.

You can (very very loosely) think of the intensity of the light being the height of the ripple, so the more intense the light the more pronounced the ripple.

Experiments have shown that it takes a fixed minimum amount of energy to set off a ripple of a given frequency. That minimum energy is given by hf, where h is a tiny number known as Planck's constant (named after Max Planck) and f is the frequency of the ripple.

You can, very loosely again, think of a photon as the tiniest ripple you can make of a given frequency. If you want to increase the intensity of the ripple you have to build it up in units of a photon. So you can think of a beam of light as being the cumulative effect of billions of tiny ripples adding together to make a bigger effect.

Photons are (again loosely) given off by charged particles, in circumstances in which the particle loses energy which is transferred into the photon ripple. The frequency of the photon is given by e/h, where e is the energy taken from the charged particle (and h is Planck's constant again).

It's not right to think of the charged particle 'vibrating', so you can't think that the particle has a frequency of vibration that's linked to the frequency of the photon, although that is a tempting image and would be in keeping with classical ideas about electric fields.

Indeed, one the reasons that made physicists realise that there was something wrong with classical electromagnetism was that they imagined electrons orbiting in atoms like tiny planets, and classic electromagnetism said that the electrons would indeed create ripples as they orbited. That would mean that the electrons would be radiating off energy all the time, so they would soon slow down and spiral into the nucleus. The early ideas of quantum theory were that the electrons could only exist in certain orbits, in which they didn't create ripples, and that the ripples only happened when an electron 'jumped' from one orbit to a lower one. The energy given off by a single jump was the minimum needed to start a ripple, or in other words to create a photon.

You don't need to go all the way into quantum mechanics to understand what's meant for "light frequency".

Classically, light is energy being transported by the electromagnetic (EM) field. When the electron vibrates it modifies the EM field in such a way that energy propagates away. This field is a vector field, that means for every point in space the EM field has 6 values defining it, 3 for the electric field and 3 for the magnetic field.

For simplicity, think only about the electric field for now. Every point in space is associated with 3 numbers wich determine the vector of the electric field there. As time goes by, those 3 numbers change because the field is dynamic so the vector that represents is constantly changing. In the case of monochromatic light this change is periodic. As time passes, the vector at any point may describe a circle, an ellipse or just get shorter and then longer like a spring. This periodic movement has some frecuency and that's what is meant by the light's frequency.

For example, red light has a frecuancy of about $4.3*10^{14}$ Hz. This means that if you could measure the electric vector in some point in space, you'd see it oscillating $4.3*10^{14}$ times every second.