Graviton and photon I understand that photon is the carrier of a change in the EM field. 
Is the graviton a carrier of the gravitational field itself or a carrier of a change in the gravitational field? 
 A: I suspect you have misunderstood what a photon is. A real photon is effectively a unit of energy exchange. Suppose you are irradiating a metal surface with light (obviously I'm referring to the photoelectric effect here) then the light can only exchange energy with the metal surface in chunks of $h\nu$ and these chunks are the photons. Likewise if you are generating a light wave then you'll find you can only put energy into the light wave in chunks of $h\nu$ and again these are the photons. When light is propagating it doesn't make sense to describe it as photons - it isn't like a hail of little bullets. We only see photons when the light is exchanging energy with something else.
Exactly the same applies to gravitons and gravitational waves. When a gravitational wave interacts with something it can only transfer energy to it in multiples of the graviton energy, or when we are generating a gravitational wave we can only add energy to the wave in multiples of the graviton energy. However the graviton energy is so small that we are unlikely ever to detect one in a collider.
When you are talking about EM fields I would guess you are thinking of virtual photons, or for gravity virtual gravitons. But these virtual particles are really just a mathematical device and they don't really exist.
A: You can say that an electromagnetic wave is a carrier of a change in the electromagnetic field. If I have a charge and move it, the acceleration required to move a charge causes an electromagnetic wave to ripple out, which you can interpret as signalling to the rest of the world that the charge has been moved.
Similarly a gravitational wave signals a change in position of a mass. Now gravity is a lot more complicated then electromagnetism, but if we stick to 'ordinary' situations where you have an isolated compact object surrounded by asymptotically flat spacetime, then there is a very close analogy between what happens in electromagnetism and what happens in gravity (meaning that the equations governing the waves of both are the same to a good approximation).
A photon is a quantized piece of energy of the electromagnetic field. It need not be associated with an electromagnetic wave. As such there is not in general a sense in which a photon is associated with a change in the electromagnetic field. For one thing you can describe repulsive forces between electrons by the exchange of virtual photons, but repulsion of point charges is associated with static fields. Second, more abstractly but also more correctly, the photon is a particle picture of electromagentism which is complementary to the field description, in the sense that you really shouldn't talk about both a field picture and a particle picture at the same time (more precisely the quantum mechanical operator that counts the number of photons does not commute with the operator that tells you the value of the electric field).
So--electromagnetic wave signals changes in electromagnetic field, and a similar statement holds for gravitational waves. BUT, associating this with photons (or gravitons) is taking things too far.
A: In the standard model of the physics of elementary particles there exist the gauge bosons of the three forces, the photon for the electromagnetic, the gluon for the strong and the W and Z bosons for the weak.

In particle physics, a gauge boson is a force carrier, a bosonic particle that carries any of the fundamental interactions of nature.1 Elementary particles, whose interactions are described by a gauge theory, interact with each other by the exchange of gauge bosons—usually as virtual particles.

Within the calculations virtual particles carry the quantum numbers of their name, but are off mass shell. The photons and the gauge bosons are also real particles in the Feynman diagrams describing interactions, and the gluons are seen as gluon jets. Thus the existence of gauge bosons is established with the data.
In a unified theory that will include gravity, the graviton will be the equivalent gauge boson for gravity. In the effective theories of gravity used in cosmological models it does play the same role for gravitational interactions that the other gauge bosons play for the their respective interactions.
If by "change" you mean "an interaction" then yes, at present the graviton plays the role of the gauge boson for the gravitational interaction. It remains to be seen whether a unified gauge  theory with quantized gravity materializes to be able to comment further.

The graviton is the exchange particle for the gravity force. Although it has not been directly observed, a number of its properties can be implied from the nature of the force. Since gravity is an inverse square force of apparently infinite range, it can be implied that the rest mass of the graviton is zero. The current experiments LIGO and VIRGO seek to detect gravitational waves, which would be considered to be coherent collections of gravitons. 

bold mine. Note that light, electromagnetic waves, are a coherent collection of photons, as the analogy.
The LIGO experiment announced the discovery of gravitational waves today, and published the results.
A: 
I understand that photon is the carrier of a change in the EM field. 

That's wrong I'm afraid. When you turn on an electromagnet there's no photons flying around. Magnets don't shine. Some will tell you that the photon is the messenger particle of electromagnetic force, and that the electron and the proton in the hydrogen atom exchange photons. That's wrong too. Hydrogen atoms don't twinkle. They're referring to virtual photons, and like John Rennie said, "these virtual particles are really just a mathematical device and they don't really exist". Anna v said the same here: "thus virtual particles exist only in the mathematics of the model".
Virtual photons are field quanta, or abstract chunks of electromagnetic field. When the electron and the proton attract one another, they "exchange field" such that the hydrogen atom has very little in the way of an electromagnetic field left. So the exchange mechanism does have a sensible foundation. But the notion that electrons and protons are forever spitting out photons and absorbing photons is popscience I'm afraid.     

Is the graviton a carrier of the gravitational field itself or a carrier of a change in the gravitational field?

Neither sounds quite right. Go back to the photon, and think of it as a singleton electromagnetic wave of energy E=hc/λ. It isn't a carrier of the electromagnetic field because it isn't something like an electromagnet, and it isn't a carrier of a change in electromagnetic field because there are no photons flying around when you switch on your electromagnet. Instead it's an electromagnetic wave, propagating at c, a transient passing change in the electromagnetic field. It isn't some step-change in the electromagnetic field. Once it's passed you by, the situation is like it was before.  
The graviton is the same thing in a gravitational context. It's a singleton gravitational wave. Only nobody knows whether there's anything akin to Planck's constant of action h for gravitational waves. If there isn't, the word graviton is arguably a misnomer, and perhaps should be discarded. 
Note that just as there are no photons flying around an electromagnet, there are no gravitons flying back and forth in the room you're in. Gravitons are not responsible for the force of gravity. Also note that when two bodies attract one another, their combined gravitational field is not reduced, so the idea of virtual gravitons in an exchange mechanism doesn't really work. Moreover the photon has a non-zero "active gravitational mass", but it isn't accompanied by a graviton. You might say it's a carrier of the gravitational field itself but I'd rather just say photons and gravitons are merely different types of waves in space. 
