Does every elementary particle have its own separate field? Higgs field is pretty simple for me to understand, you have one field that creates one particle (Higgs boson).
So I continue to assume one field one particle.
Up field creates a up quark.
Down field creates a down quark.
Strong field creates a gluon.
Electron field creates a electron.
Electromagnetic field creates a photon.
...
 A: You're right that one field basically corresponds to one fundamental particle. To be more precise, there is one type of fundamental particle for each degree of freedom (well... let me not get into that subtlety) in the fields of the standard model. What I mean by that is that you can have linear combinations of fields, but they don't correspond to brand new particles - for example, if $u(x)$ and $d(x)$ are the up and down quark fields, you can have things like $\frac{1}{\sqrt{2}}[u(x) + d(x)]$, but that doesn't make a new particle, it's just a superposition of an up quark and a down quark. This is relevant because e.g. the photon field is actually represented by a linear combination of what you might call two separate fundamental fields.
Off the top of my head, I can think of 58 quantum fields directly included in the standard model, corresponding to the following particles:


*

*(6) Left-handed and right-handed electron, muon, and tau lepton

*(3) Left-handed electron, muon, and tau neutrinos

*(36) Left-handed and right-handed quarks of six flavors (down, up, strange, charm, bottom, top) and three colors (red, green, blue)

*(4) Electroweak bosons (W+, W-, Z, photon)

*(8) Gluons of all non-singlet combinations of two of the three colors

*(1) Higgs field


I think it's considered likely that there are 3 additional right-handed neutrino fields, although that's not actually confirmed yet. Plus there's a gravity field, corresponding to the (hypothetical) graviton, but that's not part of the standard model. And of course there is the possibility of multiple Higgs fields.
In general, each field takes its name from the corresponding particle. The only exception I can think of is the photon, whose field is sometimes called the electromagnetic field. But to be accurate, the photon field actually corresponds to the electromagnetic vector potential $A^\mu$, and the thing you may be used to hearing called the "electromagnetic field" - the tensor containing $\vec{E}$ and $\vec{B}$ - is actually the derivative of that potential $A^\mu$.
A: This is the modern definition of an elementary particle--- a particle which is the quantum of a field in the renormalizable Lagrangian. There used to be other definitions, but they are no longer considered fundamental, since the elementary particles sometimes can't be isolated. This means that quarks aren't poles in the S-matrix, as particles were defined by Wigner, and neither are gluons. Also, some particles are massless, and they cannot be nonrelativistic, so they aren't objects with a straightforward forward-in-time position wavefunction description, as particles were defined in the 1930s.
In the modern definition, there is one kind of elementary particle for every field, but the correspondence you write is wrong:


*

*Photon: electromagnetic field (which is really a Higgs-induced linear combination, a mixture of a fundamental field and a part of another fundamental field)

*W/Z bosons: weak gauge field (the Z and the photon are mixed together)

*Gluon: strong gauge field (not mixed up with anything, but confined)

*quarks, u/d/c/s/t/b: each of these have a field. Technically the left parts are separate fields from the right part, but they are mixed by the Higgs into massive pairs.

*leptons e/$\nu_e$/$\mu$/$\nu_\mu$/$\tau$/$\nu_\tau$: Each of these have a field. Again, the left and right parts are fundamentally separate, but mixed by the Higgs. This time, the neutrinos are left-over after the electron-like leptons make three massive pairs.

*Higgs: in the standard model, it is an elementary scalar, only the radial part is a particle, the rest is absorbed into polarizations of the W's and Z's.


That's all the elementary particles in the standard model.
A: Note that there are fields which do not have a physical particle associated to them. These are fields that are included for technical reasons. For example,  the Faddeev-Popov ghosts fields in the SM.
Furthermore, in QFT in fixed curved backgrounds the particle concept is trickier. In dynamical backgrounds the situation is even worse.
