What do we mean by charge in physics? I am drawing the comparison between electrical charge and colour charge, in electric charges they communicate with (virtual) photon and photon itself is a boson carrying no charge. How about colour charges how do they communicate with each other if they themselves are the boson?
 A: Charge in this context refers to the fundamental coupling between particles (either matter or gauge bosons) and gauge bosons. Let me explain, starting with the QED sector:


*

*The electron is electrically charged. This means it interacts with the photon. 

*The photon is electrically neutral. This means it does not interact with other photons at tree level. (There are higher-order photon-photon interactions mediated by virtual particle-antiparticle pairs, but there is no fundamental photon-photon vertex.)


Now we get on to the QCD sector:


*

*The quarks are electrically charged. This means they interact with photons.

*The quarks are also "colourfully" charged (they have colour charge). This means they interact with gluons.

*Gluons also have colour charge. This means that (unlike photons) gluons interact with other gluons. There is a gluon-gluon interaction vertex, even at tree level. 

A: 
What do we mean by charge in physics?

A)
All words that we use in physics come from the classical  framework, observing how matter behaves.
Charge comes from the first observations of electricity, how rubbed pieces of matter attracted each other, so the word  "charge" caries  the meanings before it was identified with electricity,  take for example 1 c

to impose or record as financial obligation,
  charge debts to an estate

So by rubbing one "charged" the piece of matter.
Then mathematical modeling came in and charge became the quantity identified when fitting data with the  force between two "charged" pieces of matter , Coulomb's law.
B)
In the quantum mechanical frame this "force" comes as the $dp/dt$ of particles exchanged , the coupling constants at the vertices defining the force.
The color quantum number was invented :

The term "color" was introduced to label a property of the quarks which allowed apparently identical quarks to reside in the same particle, for example, two "up" quarks in the proton. To allow three particles to coexist and satisfy the Pauli exclusion principle, a property with three values was needed. The idea of three primary colors like red, green, and blue making white light was attractive, and language about "colorless" particles sprang up

By giving the quarks and gluons a color charge, with three possibilities the algebra becomes complicated. The gauge bosons themselves,gluons,  by carrying color charge are self attracting and the strong interaction can be modeled successfully .
So, just as quarks carry electromagnetic charge +/-1/3 or 2/3 , they also carry "color", the difference with electromagnetism is that the gauge bosons, the gluons carry color. 
You ask:

I am drawing the comparison between electrical charge and colour charge, in electric charges they communicate with (virtual) photon and photon itself is a boson carrying no charge.

At the particle level any interaction is communication, so electromagnetic interactions do not happen only with photons, but any exchange particles in the Feynman diagram. It is the coupling constant at the vertex that decides whether the interaction is weak , electromagnetic or strong.

How about colour charges how do they communicate with each other if they themselves are the boson?

Color charges are not the boson, they are quantum numbers identifying the strongly interacting particle.
Again they communicate  by interaction of exchanged virtual particles, and again it is the coupling constant at the vertex that insures if the interaction is weak , strong, or electromagnetic. It is the allowed by quantum numbers vertices, which with three colors and self interacting gluons make the color force so binding to the primary quarks so as to have stable hadrons.
