Is there a definition of electric charge and proper explanation of it?
It is said "Electric charge is the physical property of matter that causes it to experience a force when close to other electrically charged matter." How is it though that matter can get charged?
Defining charge as the property of feeling a force with other charged matter seems circular. What is charge? Is there a non-circular definition / explanation?
I think every fundamental definition is kind of going in circle.
I would say an electric charge is something that obeys Maxwell's laws. But to write those laws, you have to know $\vec{E}$ and $\vec{B}$ which need a definition of an electric charge.
At the end you just group things that look/react alike and named them. The problem arise when you have to define what they have in common. For example, you could say that a charge is something that can interact with a photon. But what's a photon? It's a particle that carry interaction between charge.
The same happens when you try to define mass as something that resist to motion set by a force to be moved and than force like something applied to massive object to make them move
I don't see a way out of this circle. That's puzzling but you just have to live with it I guess
Practically, for a macroscopic body such as a chunk of metal, the charge on that body is the difference between the number of electrons and protons in the body. It is hard to knock a proton out (can be done, though), but for conductors we can push and pull electrons out by supplying a bit of energy (back to that in a moment). However, instead of saying the a chunk of metal has one more electron than the number of protons, we say that the object has $-1.6\,10^{-19}$ Coulombs of charge. Similarly, if we have one fewer electrons, the object has $+1.6\,10^{-19}$ of charge.
Why do we care? Phenomenologically, it is because electrons and protons attract each other while two of the same type repulse each other. We can talk about underlying theories as to why, but those theories are trying to explain the physical phenomenon that we see if we pass a free electron near a free proton.
So, if we have an excess of electrons in one object and a dearth in the other, the excess electrons in the first attracts the excess protons in the other and the objects will get pulled together (e.g. capacitor plates attract each other). For two charge neutral objects, the bulk electrons in one object still get attracted to the protons in the other, but they are equally repulsed by the bulk electrons in the other and there is no net force. (Note that is is possible to generate some force between two neutral objects if the electrons aren't uniformly distributed across the object because there is distance dependence on the force).
Hopefully that helps with one part of your question. The second part was how do you charge an object. Because each and every electron in a neutral object is attracted to the protons in the object, if we were to pull and electron off the object, it will have a force pulling it back to the (now) charged object. We have to do some work and supply some energy to get that electron off. There are a lot of ways to do this, but the most common is probably a battery. A battery uses chemical processes to pull electrons from its positive terminal to its negative terminal. So the positive terminal is positively charged and the negative is negatively charged. Another way is to hit an object with high enough energy light such that the electron pops off the surface. That is how the Millikan oil drop experiment worked. Pop electrons off the drops of oil and see how the drop moves when put between two metal plates where one plate has excess electrons and the other has a dearth of electrons. The drop will move.
to be more specific about the definition of charge, charge is an intrinsic property of inherent matter. As we all know the mass which is considered as the fundamental property of every particle in this universe, electric charge is considered as the fundamental property of the particle that is used for electrostatic purposes.from the Franklin`s view he defined the electric charges as the which is used for the electrostatic interaction. Everyone knows that static charge produces an electric field and moving charges produces currents. In general when we deal quantum mechanics charge is considered as number rather than thinking about the importance of electric charge. spin is considered as the property of the atom which is hard to visualize in QM.
