I would like to know what charge actually IS. Not the 'flow of electrons' charge but the charge because of which protons and electrons attract. I want to know why these attract and what the difference is between them. Why do we put a positive on a proton and a negative on an electron? They are 'positive' and 'negative' but what is the difference?
When countless experiments pointed towards that only two types of charges exists, then what would we call them? Maybe just type 1 and type 2 for example? No, that leaves open the possibility that a type 3 might exist. Rather let's pick some binary terms. Something like plus and minus. Or day and night. Or positive and negative. It doesn't matter which is called what, we can't see the difference anyways. We just make a choice.
There is no difference between positive and negative charges. The naming could have been opposite, had history been different. The only important thing we know about them is that same types repel and different types attract. That repulsion and attraction phenomenon is hard to explain - it exists, and we don't know why. And we have chosen the name charge to describe these "things" that show this phenomenon.
I glanced through the proposed duplicates, and I want to put in the point of view of an experimentalist.
There exists an everyday word called "electricity." The root comes from the greek word for amber ηλεκτρον. Amber is a naturally found "stone", fossilized tree resin, and its property of attracting stuff was known from ancient times.
Around 585 BC, Thales discovered that if he rubbed amber (ilektron) with a piece of fur, that amber could attract lightweight objects (like feathers) to itself. Thales had discovered the principle of static electricity.
Because he lacked the tools to investigate further - as did subsequent thinkers and experimenters for more than 2,000 additional years - no one really followed-up on Thales’ ideas until the late-17th and early-18th centuries.
It is an observational fact that some matter, when rubbed, displays attraction and repulsion. This is two states, and mathematically easily described by assigning a positive sign and a negative sign to the variables eventually used to measure the observed effects.
That charge is carried by particles was found experimentally in the cathode ray tubes, and the assignment of the charges to particles follows the history of physics from then on. Consistency in assignments is important, but whether the electron was dubbed with a negative charge giving the proton a positive one is just a historical fluke.
These observations were organized into laws, which were unified in the electromagnetic theory so well modeled with the mathematics of Maxwell's equations. . The quantum mechanical framework of nature is consistent with the macroscopic observations and incorporates the effect in the mathematics.
I just want to stress that physics is about describing observations and data with mathematical models. To do that there are certain postulates, laws, principles that are assumed so that the mathematics fits the observations. The existence of two charges is one of the basic observational facts incorporated into the mathematical models of nature, the sign is arbitrary but consistent, and historically it is the electrons that are called negative.