What is "charge discreteness"? I assume it is some kind of quantity. Google only made things more confusing. 
I get that it has something to do with circuits. 
I also get what a discrete charge is. In fact, I thought charges were, by definition, discrete - because each individual proton/electron/hole contributed one unit of charge. 
 A: Charge discreteness is the statement that charge comes in packets which are of size 1 electron charge. You can understand this as a consequence of the fact that everything is made of particles with definite charge, but then it leaves the question of why the antiproton and the electron have the same charge.
All particles have opposite charge to their antiparticle, since the particle and antiparticle are always related by reversing-time and reflecting space (this is the CPT theorem). This is one precise statement of the fact that anti-particles are particles going back in time. But there is no observed process which takes a proton to an anti-electron, so the charges of the two are not obviously related.
But the charge on the proton and electron are related in three different ways in modern physics.


*

*Magnetic monopoles require charge quantization to be consistent. The vector potential is defined by the phase shift that it makes on charged particles, and it gets multiplied by the charge to turn into a phase. If the charges are quantized, then a certain amount of vector potential over a line is equivalent to nothing, so that a certain amount of magnetic flux is indistinguishable from no flux.

*In standard models of grand-unification, these monopoles catalyze proton decay. This is called the Callan-Rubikov effect.

*Black holes can convert charge in protons to charge in electrons, leaving behind the difference of the two charges. Small charged black holes are necessarily light in a string theory or quantum gravity completion, so you can conclude that if charge is not quantized, there are very light charged particles running around.


The first two points are closely related, because if you have a discrete gauge group, you can make a monopole field on the complement of a small sphere. If you have a grand-unified theory, this monopole field can extend to the interior of the sphere, so long as the full group is compact (meaning that there is no unquantized charge in the big theory). So the idea of quantized charge is ultimately tantamount to the existence of monopoles.
The other way to extend a monopole field on the complement of a sphere to the interior is to make the sphere a black hole. This links the first two cases to the third. In all modern theories of physics, charge is quantized for fundamental reasons, and magnetic monopoles necessarily exit for the same reason.
So the notion of charge quantization is really a fundamental thing--- it says that all particles we discover with electric charge will have a charge which is an integer multiple of the electron's charge (for quarks this is divided by 3--- the actual statement is for the U(1) charge of the fundamental gauge group, and the fundamental charges are integer multiples of 1/6--- all of these crazy charges are explained by an embedding in SU(5) grand unified theory).
