Im sure I'm pretty undereducated to be asking a question like this, but what do the colors represent in quark chromodynamics? A I understand it, each color represents a sort of charge, but what kind of charge; is this electromagnetism? Or is this some sort of strength of the strong force?
It is not electromagnetism, but it is a charge. For example, while there is an electric charge and this gives the strength of the effect of an electric field on a particle, there is also gravitational charge (mass) that gives the strength of the effect of the gravitational field on a massive particle. In this sense the colour charge is a charge that gives the strength of the strong field on the particle.
But, keep in mind that in terms of quantum field theory, the charge is not so much a fundamental numerical property of the particle but a more complicated matter of how it interacts with other particle fields. In effect, two electrons interact indirectly by both interacting with photons. And usually when it comes to describing interactions between quarks, it is this type of logic that is being used, rather than the idea of a field of force - in the mode of Maxwell electromagnetism or Newton gravity.