In what way are the qualities of mass and charge related?
qualities? well, mass and charge are both measures of properties that determine how forces will act upon the objects that have a non-zero quantity of them. These forces both happen to be 1/r^2 forces.
I’m not sure exactly what you’re asking, so I’ll guess.
I’m not a physicist but, here goes my meager knowledge.
Charge isn’t a matter of mass. It’s a matter of free electrons or lack thereof at the atomic level. If the number of electrons and protons is equal, the charge polarity will be neutral. Extra electrons will give a negative charge, etc. In a molecule, electrons are often shared between atoms orbits. The end count between protons and electrons should still be the deciding factor of charge polarity. Now, obviously the number of free atoms or lack of them increases the potential for the exchange of free electrons or charged ions with an oppositely charged mass. This is commonly expressed as voltage + or -. In solids, free electrons are the general means of transfer, while in liquid solutions, such as salt water, matter may travel in either direction.
If you are referring to gravitation, no one is 100% sure what exactly causes it. The best theory was given by Einstein, and it’s nowhere near simple. As a matter of fact, we have yet to solidly prove or disprove it eighty years later! Spacetime and matter are directly related. In other words matter affects spacetime. It causes it to curve or warp. The resulting distortion funnels other matter towards it much like a vacuum attracts air or a depression in the ground gathers water running by it. The resultant distortion is relative to the mass. In other words the distance the gavitational field extends is dependant upon the size and density of the mass. This attraction affects everything, even light!
Fascinating stuff! To bad you have to be an accomplished mathmetician to fully understand or appreciate it.
No, GR correctly describes gravity as well as we can measure it. It does an extraordinarily good job at it, too. Evidence for its correctness abounds as it explains all observed gravitational phenomena to date (with some possible fringe exceptions that are not necessarily germane to the paradigmatic discussion). How much more solid do you want?
By the by, mass goes a bit beyond simply gravity as it is also (in non-relativisitic terms) what dictates how much acceleration an object has when any given force is applied to it. As far as we know, these two definitions for mass are equivalent.
At the simplest level, the electrostatic and gravitational forces work in a very similar manner. The basic equation for electrostatics is Coulomb’s law, F = k[sub]C[/sub]q[sub]1[/sub]q[sub]2[/sub]/r[sup]2[/sup] , where F is force, k[sub]C[/sub] is a constant of the Universe, and the qs are two charges. For gravity, it’s Newton’s law, F = -Gm[sub]1[/sub]m[sub]2[/sub]/r[sup]2[/sup], where G is a constant, and the ms are mass. Notice the similarity. The biggest difference, at first glance, is the negative sign, which means that like masses attract, but unlike charges attract.
The similarity ends if you look too deeply, though. The gravitational field itself can act as “mass” to produce its own field, leading to the more complete theory of gravity, Einstein’s General Relativity. Electromagnetism doesn’t work this way: The electric field does not have a charge. Also due partly to this complication of gravity, there is no quantum theory of gravity, but there is a very good quantum theory of electromagnetism. If you try to use the same methods for gravity, you end up with nonsensical answers.
Aha! Some brains well versed in this stuff! Keep going, don’t let me stop you. My expertise doesn’t go far beyond Discovery channel specials and the yellow book “GR for dummies”! 
Hmmmm, can’t edit posts here. I’ll have to keep a sharper eye for typos from now on before submitting the post.
There’s one other important relationship between mass and charge:
To date, no charged particle has been discovered that does not also have mass. (Massless “particles”, such as photons and neutrinos* and gluons, are electrostatically neutral.)
[sub]*) I just know I’m going to get lynched for saying that neutrinos don’t have mass. So lynch me.[/sub]
Are mass and charge separate external manifestaions of a common internal quality?