What is the *quantum* of a field? The particles of nature are the quanta of relativistic quantum fields, from what I've understood.
But what does this mean physically?
Is the electron the quantum of an electron field? In what sense, in practice?
Is there only one electron field in the universe or is there one for each electron?
 A: To gain some insight, first consider the quantum harmonic oscillator which has the nice property that states of definite energy have energy
$$E_n = \left(n + \frac{1}{2}\right)\hbar \omega\,,\; n = 0,1,2,... $$
So, it makes sense to identify a quantum of energy as $\hbar \omega$ and then the quantum number $n$ is the number of energy quanta present.
Now, consider the classical field that obeys the wave equation.  By separation of variables, we can identify the the modes of the field and find that each mode obeys a harmonic oscillator equation of motion.
So, we can express the Hamiltonian of the field as a 'sum' (integral) of an infinity of independent harmonic oscillator Hamiltonians.
Finally, we quantize the field by expressing the Hamiltonian as a 'sum' (integral) of independent quantum harmonic oscillator Hamiltonians, each with a different energy quantum.
Then, for each mode, there is a quantum number $n(\mathbf k)$ that is the number of energy quanta present.  On this view, the particles are the mode quanta of the quantized field.  If all of the modes are in their ground state, there are no particles present and this state is the vacuum state.
Thus, there is indeed just one electron 'field' with electrons (and positrons) as the mode quanta.
What does this mean physically?  The ontology of QFT is still unclear.
Lastly, all of the above is in the context of free (non-interacting) quantum fields.
In order to get interactions, there must be non-linear terms involving the field with itself of the field and other fields.  However, the non-linear terms somewhat spoil the particle interpretation outlined above.
