Cibek said:
I don't mean to offend either of you, but all of this sounds to me that we just haven't found out why yet, and the only explanation that we have so far is that "It is that way, don't ask about it".
That's certainly possible, even probable--almost no one believes that the standard model of particle physics is complete as it stands. For any given question, it's always possible to keep asking "why?" until you reach the boundary of our knowledge about the universe. That said, we have today a very nice predictive and self-consistent understanding of electricity and magnetism. While it may not be Ultimate Truth, there aren't currently any particular loose ends in this part of physics that you could pull on in the hope of unraveling something more fundamental. So our current theories postulate the existence of electric charge without explanation and leave it at that.
Cibek said:
You say that the electron has the negative charge, and that it's a basic part of nature. My question is then, what about anti-protons? They have a negative charge, and what is it that gives them that charge and what makes them different from matter-protons?
Protons are made of two up quarks with charge +2/3 and one down quark with charge -1/3 for a total charge of 2/3 + 2/3 - 1/3 = +1 (in particle physicists' units of charge). Quantum field theory predicts, for any particle, the existence of an antiparticle which has opposite charge. This fact is well understood as a deep mathematical consequence of the union of quantum mechanics and special relativity. Thus there is an antiproton composed of two up antiquarks each with charge -2/3 and one down antiquark with charge +1/3, for a total charge of -1. So the charge of the antiproton, and its difference from the proton, is well understood.
Cibek said:
Could it be that we just haven't discovered yet that both quarks and electrons are divisable? Maybe they are composed of even smaller parts that gives them their properties?
Certainly this is possible. Experiments are always testing this possibility. So far we have zero evidence that either quarks or electrons are composite particles as opposed to fundamental ones. That of course does not rule out compositeness, since our experiments can only probe down to a certain length scale. But in the absence of evidence otherwise, the theories we construct to explain our data postulate electrons and quarks as fundamental particles.
Now, one may try to explain the otherwise inexplicable properties of the known particles by postulating that they are somehow composed of something more fundamental. This is a valid idea, but only useful if the new "deeper" theory explains more facts than it makes assumptions. As a trivial example, if a proposed theory explains why all electric charge comes in units of a fundamental charge, but assumes the existence of "supercharge," which only comes in units a fundamental supercharge, you haven't really gained much, especially if your theory makes no other testable predictions besides reproducing the known data, which is already explained by the standard model. And it turns out to be hard to explain the current data with a theory simpler than the standard model. So in particular we have no reason to believe that, say, electrons are composite.
String theory is an example of a more fundamental theory that might explain things like the charges of the known particles in terms of more fundamental objects. Of course, string theory must itself make certain postulates about the existence and behavior of strings. The hope is that these postulates end up being more concise than the mess of particles we currently need to postulate in the standard model.