Why, fundamentally, are particles charged? This is something that has long bothered me, and I have asked a few physicists and chemists and never gotten a very satisfying answer. Why are particles charged? And I'm not asking (and this is the answer you often get) why some molecule might be negatively charged because the answer to that is simply is that there was one more negatively charged particle than positively charged particle. That doesn't answer my question, however, of why there are charged particles at all?
So, on a fundamental level, what makes one particle charged and another one neutral.
For instance, protons and neutrons have almost exactly the same mass, which I'm assuming that means they have very similar internal compositions, yet one is positively charged and the other neutral. What is going on that makes these particles different?
On a similar note, and I'd really like to know this, how is it that a proton and an electron have exactly equal and opposite charge, yet an electron is about 1000 times less massive than a proton?
I have particular trouble with this question because if the answer were something along the lines of "particle X which composes protons all have partial positive charges of n amount..." I would just ask why particle X has any charge and we'd be in the same predicament.
I'm sure there is a satisfactory answer to my question, I just can't find it and don't know what resources are reliable or detailed enough to be satisfactory.
 A: 
Why, fundamentally, are particles charged?

The answer to this and the similar questions is : because that is what we have observed and defined by measurements with innumerable data. It is an existential question. Physics is not about existential questions.
Physics is about observations with measurements of the way the natural world behaves and fitting with mathematical models the observed behavior. The models are successful, validated, when they can predict any future behavior accurately.
Starting with the Millikan oil drop experiments the electron was found to have a definite charge and the microcosm of elementary particles made mathematical sense , from nuclear physics to quark physics with assigned charges and masses and a number of quantum numbers. The ingenuity of physicists found that the standard model, a mathematical model ,  could describe all interactions ( almost) of elementary particles and their composites with great accuracy.
Once a mathematical model is successful many people , mainly theorists, tend to agree with a platonic philosophy ( mathematics creates nature), and that is why  your "why" question gets all these explanations from symmetries etc. But the physics truth is that a mathematical model is a tool, not a creator of nature. If we had had different observations we would have had different mathematical models.
IMO physics answers "why" questions by "how" using mathematical models and postulates that relate the mathematics to physical observables, one fits the observations. Why this is so? Because.
A: I am not sure how to answer all of your question but I can answer the neutron and proton part. The charge difference is due to their composition. A neutron is composed of two down quarks and one up quark. A proton is composed of two up quarks and one down. Up quarks have a charge of +2/3 and down quarks have a charge of -1/3. A neutron with composition udd has a charge of 2/3 - 1/3 - 1/3 = 0. A proton with composition uud has charge of 2/3 + 2/3 - 1/3 = +1.
A: This is ultimately a very deep question. We do not have a very easy ability to answer it at this time. Let me give you the pieces that we presently have.
The basic interactions.
The world as we know it today consists of five fundamental things that happen; they are all called "fields" and the first four are called "forces" or "interactions". In rough order from strongest to weakest, these 4 are as follows. 


*

*Protons and neutrons are sticky and stick together into atomic nuclei. This is called the strong nuclear force or the strong interaction. Some matter does not feel the strong force; these particles are called "leptons." Some matter does, and they are called "quarks" (and combinations of them are called "hadrons"). 

*Radioactivity. Neutrons want to turn into a proton-plus-electron-plus-antineutrino triad. In particle physics this is known as the weak nuclear force, or the weak interaction.

*Chemistry. Protons attract electrons and repel protons; electrons repel each other too; they emit light under certain conditions. To particle physics, this is all known as "electromagnetism."

*Things fall down. Matter mysteriously attracts other matter through an unscreenable infinite-range force known as "gravity".


In those four interactions, and mostly in that third one, we think we can find mostly everything that happens in this world. The fifth interaction which is conspicuously absent is the "Higgs field", which we'll talk about in a moment.
We cannot give you a great explanation why Nature hasn't chosen fewer interactions or more interactions; in particular we cannot tell you a deep reason why Nature chose for there to be an electromagnetic interaction at all. We can give you details but you can always persist asking "why" until we find ourselves in ignorance.
And then there was unity.
We can tell you some other things about electric charge though. A full classical theory was available per James Clerk Maxwell, and a full quantum theory became available due to the works of Schwinger, Tomonaga, and Feynman. This became a template for understanding the strong force, with photons replaced by "gluons" and electric charge replaced with three "color charges" (called arbitrarily "red", "green", and "blue") which has to balance out either in the same way that electric charge does (quarks plus antiquarks) or else by having even amounts of the three charges.
In particular, physicists beat their heads against a wall trying to describe the weak interaction in a way that (a) didn't have unphysical consequences and (b) played nice with quantum relativity until 1961, when a young Ph.D. named Sheldon Glashow discovered that you could make relativistic quantum theory and the weak interaction play nice together if you "bundle in" the electromagnetic force and describe them both together, mediated by four types of 'virtual' particles: two of those particles, the Z boson of the weak interaction and the photon of electromagnetism, at very high energies can phase into each other. 
Glashow's "electroweak" theory had some unphysical consequences at high energies, which were removed in the mid 60s by postulating this fifth fundamental interaction, which everyone calls the Higgs field. The Higgs field has a weak coupling to all of the "matter" particles but also to some of the "force" particles, giving them mass. In short, the Higgs explains why everything which is not a proton or a neutron doesn't zip off at the speed of light. We believe these days that we've definitively observed some statistical effects of the distinctive particle which quantizes the Higgs field, shedding most of our doubt about this explanation. The interaction-energy that these particles have due to interacting with the Higgs field grants them an effective mass via $E = mc^2$. 
Does this habit of Nature unifying these forces go further? Maybe. So-called "grand unified theories" unify the electroweak interaction with the strong interaction, but they often have some strange predictions, like magnetic monopoles or that protons will eventually decay into positrons-plus-photons, which we've never actually seen happen. So it's very hard to endorse these theories at this stage. So-called "theories of everything" incorporate gravity as well.
However, here we're at a significant experimental loss: we simply don't have enough experimental confirmation to decisively say "yes, this grand-unified-theory is good; no, that theory is bad."
What the electroweak theory tells us about charge.
Okay, now that you know that the electromagnetic force is a low-temperature part of this electroweak theory, what does that tell us about electric charge? It says that electric charge comes about as part of two conserved quantities, called "weak isospin" or $T_3$ and "weak hypercharge" or $Y_W.$ The quarks, for some reason, have a weak hypercharge of +1/3. The leptons, for some reason, have weak hypercharge -1. Some things, in addition, have weak isospin +1/2, and some things have weak isospin -1/2, in the same units. What are those reasons? I don't think we really know. (Also don't get too attached to the pluses and minuses; there are 4 "antiparticles" for each of these particles that have the opposite sign for all of them.)
Electric charge in these units is $q(T_3, Y_W) = T_3 + \frac{Y_W}{2}$, leading to four classes of particles: the non-electromagnetic non-strong neutrinos $q(+ \frac 12, -1) = 0$, the electromagnetic non-strong electrons $q(-\frac 12, -1) = -1$, and the electromagnetic strong quarks "up" $q(+\frac 12, +\frac 13) = \frac 23$ and "down" $q(-\frac 12, +\frac 13) = -\frac 13.$
So for example a neutron is made of two downs and an up, total charge $2\cdot\frac{-1}{3} + \frac 23 = 0.$ The weak force will sometimes turn one of those downs into an up plus a $W^-$ boson $q(-1, 0) = -1$ , which then turns into an electron $q(-\frac 12, -1) = -1$ plus an electron antineutrino $q(-\frac 12, +1) = 0.$ The two ups and a down quark will then make up a proton $\frac{-1}{3} + 2\cdot\frac 23 = +1.$
Particles, in short, are charged because they have this intrinsic weak-isospin and this intrinsic weak-hypercharge which they cannot shed. We do not know why they have these exact parameters, except that all of the possibilities are fully represented. 
Mass discrepancies.
We've secretly also answered why protons and neutrons have a mass so much higher than the electron: the electron is a single particle which does not feel the strong force; the protons and neutrons are made up of three quarks which are bound together by the strong force. (In general that is a property of the strong force: it is caused by a powerful three-way charge that we call a "color charge"; the color charge has to balance out either by having three particles which each have one the colors, or by having a particle and an antiparticle bound together with the same color charge; if it's not balanced out it will tear particles made with the other weaker forces apart until it is satisfied because it is so strong.) Satisfying the associated strong force comes with a binding energy $E$ which acts like a mass via $E = mc^2.$ 
Neutrinos and electrons, which don't feel this energy, only get a little bit of mass from their comparatively weak couplings to the Higgs field: up and down quarks also have a little mass from this mechanism. This is much smaller than the binding energy however; it is maybe 1% if the proton/neutron's mass, most of which comes from this strong force.
So that's the tip of the gory details that go into particle-physics. I don't think we can tell you yet why those particular weak hypercharges are associated with having color charge (feeling the strong force) or not; we wouldn't know until we could reliably test the correctness of grand-unified theories and determine which one our electroweak interaction is embedded in, if there even is one. (Nature might just say, "no, that's just the way I roll, no further questions.") In some GUTs, weak hypercharge is totally separate from color charge; in others part of weak hypercharge unifies with color charge at high-enough-energies.
A: The standard model of particle physics is built on gauge theory, which is a theory of local symmetries. This means that the symmetries act differently at different points in space (and time). For example, if you move everything in the universe 1m in one direction, nothing is changed. This is a global symmetry, because everything has been moved by the same amount in the same direction - in other words, it acts the same at every point.
A gauge symmetry is different, because it can act differently at different points. Consider rotating an object by a different amount at each point; it would clearly change the physics of object. In the equations of the standard model, however, you can, in a way,  rotate things by a different amount at each point. This isn't an actual physical rotation, but more a rotation of the equations. Imagine that each equation contains an angle, $\theta(x)$, that varies from point to point in space, but the physics doesn't actually depend in $\theta(x)$ in any way i.e $\theta(x)$ is completely redundant.
There is a mathematical theorem, Noether's theorem, saying that each continuous symmetry of a physical system leads to a conserved quantity, and the fact that we can choose $\theta(x)$ as we please means that we have come sort of conserved quantity. It turns out that the theory of this rotational symmetry leads to certain particles having an electric charge, which is conserved. 
So your question ultimately comes down to symmetry.
In short, symmetry = conserved quantity.
