The way I've understood it is, particles are localized excitations of fields. You know of the Higgs boson. It is the excitation of the Higgs field, which exists through all space. Similarly, an electron is an excitation of the electron field (not electromagnetic field), which exists through all space. Every single electron is a localized excitation of the same, single field.
I don't know how to answer the relativistic questions of fields yet. For that one would need to learn a lot of Quantum Field Theory.
For the layman, however, these two links are very informative:
I understand that, but is it just a neat mathematical model to treat particles as excitations of fields because it makes calculations especially easy or is there really a field out there?
My intuition for it is that there are fields at all points in spacetime (the fields of the standard model: a tiny vector of sorts), and excitations - certain patterns in these fields - behave as particles.
Combinations of excitations can be interpreted as different excitations, and by 'can be interpreted' I really mean "can spontaneously transmute into" according to the probabilities of quantum mechanics. Since all interactions occur in discrete units of these fields (except for the photon field), referring to these units as particles is a convenient and compelling approximation - but it doesn't tell the full story.
Basically, it very much seems that fields are the more fundamental concept.
Depends, if you are a realist, then there is a field; if you are a anti-realist then the electron is not real but only a convenient shorthand in the first place.
I don't know how to answer the relativistic questions of fields yet. For that one would need to learn a lot of Quantum Field Theory.
For the layman, however, these two links are very informative:
http://www.symmetrymagazine.org/article/july-2013/real-talk-... http://www.pbs.org/wgbh/nova/blogs/physics/2013/08/the-good-...