special relativity - Nature of Fields in QFT

I'm not exactly an expert in quantum physics, but this seems to be a simple question, and I can't find an answer anywhere!

There are specific types of fields used in physics: scalar fields (i.e. as in the case of the Higgs boson), vector fields (i.e. as in magnetic fields), tensor fields (i.e. as in general relativity), etc. But what types of fields are used in QFT to model elementary particles? Is my confusion simply a result of me thinking in purely classical terms?

Answer

Yes, your confusion is wholly caused by you thinking classically ;)

In a hand-wavy way, particles are certain localized excitations of the quantized fields.

The QFT picture contains the particle picture in the perturbative approach known as Feynman diagrams (and, relatedly, the LSZ formalism). There, we are given the action of our theory dependent on some fields (be they scalar, fermion, vector, Rarita-Schwinger, tensor or even higher spin). An instructive model is so-called $\phi^4$-theory with the (here, massless) action

$$ S[\phi] := \frac{1}{2}\partial_\mu \phi \partial^\mu \phi - \frac{\lambda}{4!}\phi^4 $$

Particles are obtained in the asymptotic past and future ($t = \pm \infty$) by assuming that the interaction term $\frac{\lambda}{4!}\phi^4$ does not play a role when excitations of the fields are far apart, so we have a free theory there, with free action $S_0[\phi] = \frac{1}{2}\partial_\mu \phi \partial^\mu \phi$ and the classical e.o.m. allows the usual mode expansion of $\phi$ into creation and annihilation operators of particles of definite momentum, $a^\dagger(\vec p)$ resp. $a(\vec p)$. The creators/annihilators correspond to the same ladder operators in the quantum harmonic oscillator, for example, which is why one says that they represent excitations of the quantum field. Now, the Feynman diagrams/LSZ formalism tell you what happens with what probability when you let these free particles interact - they let you calculate the scattering amplitudes, which are essentially the entries in the S-matrix. The "Feynman rules" for writing down diagrams tell us that, for our $\phi^4$ action, we have as building blocks one kind of lines/particles corresponding to the scalar field $\phi$, and that only those graphs are valid which either only include these lines not crossing at all, or those that contain vertices corresponding to the interaction term $\frac{\lambda}{4!}\phi^4$, i.e. crossings of four of these scalar lines.

Now, we can also speak of virtual particles, of which the only uncontroversial thing to say is that they are internal lines in the Feynman diagrams, which do not correspond to the real particles in our free creation/annihilation spaces, but are often spoken of as particles, as well.

There are also resonances, about which I seem to recall a suitable treatment in Srednicki, but I'm not confident in proclaiming anything about them except that they are also often lumped in under the term "particles".