What is a charge? If mass is the measure of the inertia of a body, then what is a charge?
Everyone tells that it is a fundamental property but i believe it must have a unique definition. Anyone know in this regard?
 A: Mass is slightly more complicated than you've stated: mass is a measure of both an object's inertia and its tendency to interact with other objects via gravity.  It's the case in general relativity that an object's gravitational mass and its inertial mass are exactly the same, because inertia and gravity are both properties of the interaction between matter and spacetime. Vigorous searches for violations of this equivalence principle have turned up no evidence to the contrary.
In that second sense, mass is a sort of "gravitational charge": two objects with masses $m_1, m_2$ at rest and separated by a distance $r$ have a gravitational interaction energy
$$
U_\text{grav} = -G \frac{m_1 m_2}r
$$
which is proportional to the product of the masses.
It turns out that gravity is not the only way that objects can interact with each other.  Objects at rest separated by a distance $r$ also have an interaction energy
$$
U_\text{elec} = \alpha \hbar c \frac{q_1 q_2}r
$$
where the dimensionless "fine structure constant" is given by $\alpha \approx 10^{-2}$, the constant $\hbar c \approx \rm 200\,MeV\,fm$ corresponds to a tiny amount of energy even at tiny distances, and we have a funny empirical fact that every observed value of $q_i$ is an integer.  For historical reasons (there was no reason to suspect $q$ was always an integer) we define "electric charge" of an object as $qe$ rather than $q$, with $e^2 = 4\pi\epsilon_0\alpha\hbar c$ having some other historically-interesting constants involved and an annoyingly small value in macroscopic units.
We can actually re-write the gravitational interaction the same way,
$$
U_\text{grav} = -\frac{\hbar c}{M_P^2} \frac{ m_1 m_2 }r.
$$
However the ratio between the mass of a typical lump of matter --- a proton --- and the "scale mass" for gravity, $M_P = \sqrt{\hbar c/G}$, is about $10^{-19}$, which suggests this is not a useful direction to go in.
There is no evidence that the masses $m$ of common particles are small integer multiples of anything, so that isn't a simplifying approximation in the same way.
We find ourselves living in a world where there are a fairly small number of building blocks for ordinary matter, each with a different ratio of charge to mass $q/m$, and no obvious pattern among them.
There are two other interactions that we know about which are governed the same sort of way.  One is a "strong" interaction with $\alpha_S \approx 0.1$ and charge that comes in integer lumps --- three types of "color" charges for matter, and anti-colors for the antimatter, with the possibility of making "uncharged" objects by combining all the colors.  Another is a "weak" interaction with $\alpha_W \approx 10^{-4}$ and approximately integer lumps of charge, for a complicated and beautiful reason which is too much for your question.  (Experts: the weak charge of the neutron is 1, and the weak charge of the proton is $1-4\sin^2\theta_W$.)
So there's a long-winded definition with some examples for you: a "charge" is a property of matter which describes the strength of its interaction with other matter via different features of the vacuum.  So far as anyone knows there are four of these interactions.
A: A charge has a lot more properties than only the one you mentioned.
The distinctive feature is the electric field by which a charged particle is surrounded. It is a convention that electrons negativ charged and protons positive charged. Charged particles could be separated into negativ and positive charged particles. In rigid bodies it is possible only to move a more or a less of electrons inside one body. So a capacitor has two foils (plates), separated by an isolating gap or material. Between the plates the separated charges produce a common electric field. Using a capacitor with extended distances it is possible to use them as a measurement instrument. Putting a particle between the charged plates, one will observe a movement to one of the plates (in one direction for negative charged particles and the other direction for positive charged particles) or nothing for uncharged particles. The charge of electrons and protons as well as of their anti-particles are intrinsic properties. This means that we believe that they are exist without any influence by external fileds.
Charges have the property of mass, of magnetic dipole moment and of intrinsic spin. The last two properties are a unit. Moving a charged particle (electron, positron, proton, ...) through a magnetic field leads to the deflection of the particle from its straight path. This happens if the direction of the magnetic field is not parallel to the movement of the charges or if the magnetic field is moving in space (again non-parallel to the moving charge) or the magnetic field is varying in strength.
This bring us to the last property of charges. They able to absorb and re-emit photons. During straight and positive accelerations due to an observer they absorb the energy of photons (but re-emissioning photons of lower energy). During acceleration in circle or spirals and during breaks above all they emit photons.
A: Assuming by charge, you mean electric charge,
According to Quantum Electrodynamics, (which is the most successful theory of Nature,) the fundamentality nature of the electric charge, (as a quantum number,) is due to the fact that it represents the coupling 'constant' between matter field and electromagnetic field. 
