Why is an electron negatively charged, and what is the difference between negative and positive charges? Nobody has yet defined the actual meaning of a charge, or why a negative charge is different from a positive charge.


*

*Everybody knows that positive charge is due to protons and negative charge is due to electrons, but what does the charge mean? 

*Why were negative and positive charges so designated? 

*Was it also a possibility to call the charge of an electron positive and the charge of a proton negative?
 A: In quantum theory of elementary particles (in a sense of irreducible representation of Poincare group with mass $m$ and spin (helicity) $s$) if some operator $\hat{Q}$ of internal symmetry commutes (like electric charge charge) with Hamiltonian $\hat{H}$ of given field theory, there must be
$$
[\hat{Q},\hat{\varphi}^{\dagger}_{A}(\mathbf p)] | \rangle = q_{A}\hat{\varphi}^{\dagger}_{A}(\mathbf p), \quad [\hat{Q},\hat{\varphi}_{A}(\mathbf p)] | \rangle = -q_{A}\hat{\varphi}_{A}(\mathbf p).
$$
Here $\hat{\varphi}^{\dagger}(x)$ is a field which creates particles while $\hat{\varphi}(x)$ is a field which creates antiparticles. 
So you can see two facts: the particle has charge which is equal to the charge of corresponding antiparticle with minus sign; the setting of sign of particle's charge is formal. We can set the positron charge as negative, the physics doesn't change. 
Charge is only the measure of interaction. Let's see this on the simple example. 
We know that there is interaction between two electrons which can't be described as gravitational (they are repelled). Also we know that proton and electron are attracted (this interaction also can't be described as gravitational because it's stronger than it). Then we know that this interaction force at the simplest case coinsides with $\frac{1}{r^{2}}$ law, and the measure of interaction is constant in time. We need to build the theory with these properties. The charge of given particle, as (for simplicity) the mass in case of gravitational interaction, says us about its taking part in interaction.
A: If we re-designate all positive electric charges as negative and vice versa, while keeping their absolute value, the resulting physics would be the same. So exact choice is merely a matter of convention.
It only matters that electric charge of proton and electron are opposite ($Q_\text{proton}=-Q_\text{electron}$). The charge is additive, meaning that if you have one system with charge $Q_1$ and another system with charge $Q_2$, the combined system has the charge $Q_1+Q_2$ (sum of those charges). 
Charge of electron is opposite to that of proton so combination of equal number of protons and electrons has total charge zero, i.e. it is electrically neutral. One such combination (up to additional other particles (neutrons)) is the usual (not ionized) atom.
A: The concept of electric charge is introduced to explain experiments (originally from static electricity). It is found that only two types of charges are necessary and to distinguish them and to distinguish between they are given labels. The most convenient label is positive and negative (that has some mathematical advantages). It is pure convention that protons are assigned a positive charge and electrons are assigned as negative.
It is found that all charges of the same type repel each other, while charges of different types attract each other.
A: All particles seems to be grouped under two distinct polarities based on the manner of attraction or repulsion. Those particles repelling one another are said to have like charges. Those that attract one another have different charges. Being a positive or negative charge, is a matter of convention already accepted by world scientific community. 
A Charge is a property possessed by a body when there is internal imbalance of number of particles carrying different polarities based on attraction or repulsion.
