I understand that protons are attracted to electrons and repelled by other protons (and vise versa). Why do they do that? I have had trouble finding a straightforward answer to this question, thank you for your time.
Because for whatever reason in the physics of the universe (or at least the section that we experience) matching electromagnetic charges repel each. “Why” isn’t a question that we have an answer to.
Electrons and protons have an inherent physical property called electric charge. An electron and a proton have the same magnitude of charge, but the opposite sign (negative and positive, respectively). Finally, like charges repel each other and unlike charges attract each other.
More detail about charge here:
With that said, I didn’t really answer your “why” question. This is trickier to answer. The short version is because that is the way our universe is. This also gets into what’s called the anthropic principle, which basically states that if physical constants (like the fundamental magnitude of elementary charge, for example) were different, you would not have stable matter and therefore would not have us around to ask the question.
Also, FWIW, the attraction and repulsion of opposite and like electric charges, respectively, is closely related to magnetism and is in fact the same thing. Electromagnetism is the basis of how all electric motors work. Permanent magnets are just a special case – simply pieces of an appropriate material wherein the magnetic dipole moments of some of the electrons are aligned in a particular direction rather than being random. But the attraction and repulsion of ordinary magnets are due to the same electromagnetic force, one of the four fundamental forces of the universe.
And whenever we are able to come up with an answer for “Why,” we can again ask “Why” after the answer is given. It’s infinite regress. Turtles all the way down.
One hypothesis is that us humans will eventually reach a point where we will never be able to answer a “Why.” Not because there isn’t an answer. But because our monkey brains can’t comprehend the answer.
What I don’t understand is what about protons and electrons gives them that charge?
We still have no idea what protons and electrons are. We used to view them as particles, sometimes behaving as waves. It is now obvious that they are neither, or both, or something else.
Okay, thank you. I thought it might have had to do with quarks or something like that.
Protons are made of quarks, but electrons are fundamental particles themselves and are not made up of smaller components like quarks. So whatever gives those particles charge it is not a quark in the case of the electron.
That is a pretty fundamental question that we really can’t answer.
It’s akin to asking why matter has mass.
Like mass, electric charge is an intrinsic property of electrons and protons (and quarks).
I meant as in the lack of quarks (and neutrons and protons are made of different quarks).
Actually, the same quarks, just different quantities of each type.
A proton is made up of two up quarks and one down quark (plus the binding forces between them that make up almost all of the proton’s mass).
A neutron is make up of one up quark and two down quarks (plus the binding forces holding them together).
Thanks for reminding me. What I was attempting to say is that that’s why they’re different.
Poor Feynman got a Nobel Prize for his work on quantum electrodynamics, and today people are pretending we do not know anything about it? 
Particles like electrons can interact with each other by exchanging gauge bosons (i.e., photons), and you can compute the potential corresponding to the scattering of two electrons, which results in Coulomb’s Law.
Now, why particles have electrical charge in the first place (among other things) starts to be more mysterious— why do these symmetries exist in the universe? Electromagnetism is associated to a U(1) symmetry.
For protons it does. They’re made of 2 up quarks (+ 2/3 charge) and one down quark (- 1/3rd charge). Add them up, and you get +1 charge.
Electrons are a fundamental particle. They have -1 charge just because Jesus made them that way, the same reason quarks have the charges they do.
It is my understanding that QED theory explains how charged particles interact with each other via the exchange of photons, but does it also explain the fundamental nature of electric charge (why it exists, why it has the specific value it does, or why positive and negative charges exist as they do) or the origin of electric charge itself?
(I am fairly certain that the answer to these questions is “no”, that QED theory takes electric charge as a given, that is, a pre-existing, fundamental property of matter, but the theory does not not seek to explain the nature of charge itself.)
ETA: I see you added a third paragraph to your post while I was typing that acknowledges this.
The so-called Standard Model posits a U(1) × SU(2) × SU(3) gauge symmetry, from which electromagnetism emerges in some tricky way (like electroweak symmetry breaking), but why those symmetries, and why the values of various parameters are what they are— you got me.
PS on Wkipedia there is a table of parameters including a few relevant coupling constants, and you can perform experiments to measure their values, but that does not answer the question
@pasta has an explanation for how symmetry breaking causes forces here… I don’t understand it. Here’s an excerpt
This is true for all the fundamental forces in the Standard Model. The electromagnetic, weak, and strong forces are not put into the theory, but rather they emerge from it in an elegant fashion by requiring certain symmetries of nature to be present. The relevant symmetries are somewhat subtle and mathematical in their description, but in their own way they are simple. These are known as “gauge symmetries”. These gauge symmetries insist that the laws of nature should be unchanged if you do a certain type of mathematical operation to the fields.
I mean, you could say that the charge is what it is about protons and electrons that makes them behave that way.
From what I gather there, one should keep in mind the very general principle that, according to Noether’s Theorem, if the physical action is invariant under a symmetry transformation, there will be a corresponding conserved current you can calculate.
Teasing out what the precise symmetries are in the case of the electromagnetic or weak forces and the Standard Model is somewhat “subtle”, as explained in that link; the technical aspects of it would be covered in any textbook or lecture notes even if that does not answer all the philosophical questions.