What is the difference between gravitons and gravitational waves? What is the difference between gravitons and gravitational waves? So as I understand, gravitational waves are predicted by and part of the theory of general relativity, whereas gravitons are predicted by the standard model.
We have indirect evidence for gravitational waves? So it seems that they probably exist--but what about gravitons? How are gravitons related to gravitational waves? Are gravitons and gravitational waves just two different ways of talking about the same thing?
 A: 
In physics, gravitational waves are ripples in the curvature of space-time which propagate as waves, travelling outward from the source. Predicted in 19161 by Albert Einstein to exist on the basis of his theory of general relativity,3 gravitational waves theoretically transport energy as gravitational radiation. Sources of detectable gravitational waves could possibly include binary star systems composed of white dwarfs, neutron stars, or black holes. The existence of gravitational waves is a possible consequence of the Lorentz invariance of general relativity since it brings the concept of a limiting speed of propagation of the physical interactions with it. 

Gravitational* waves have not been directly observed, experiments are designed trying to detect them. Since they are predicted from the General Relativity theory there is trust that they exist and the difficulty comes from the very weak gravitational interaction. General Relativity is a classical theory.
As pointed out in the comments to the question, electrodynamics as formulated with Maxwell's equations is also a classical theory, and describes electromagnetic waves, light being one part of the spectrum.
The underlying framework of nature though is quantum mechanical. The quantization of electrodynamics QED is a successful theory which has the photons as the carrier of the electromagnetic interactions and the building stones of the classical electromagnetic wave. It can be shown that the classical wave emerges from innumerable photons with energy $=h\nu,$ $\nu$ the frequency of the classical wave.
Gravitons are to gravitational waves the theoretical analogue of photons for electromagnetic waves. They are the proposed carriers of the gravitational interactions at the quantum level, and are expected to appear naturally in a future theory of quantized gravity. At the moment quantization of gravity can be accommodated in string theories, which are  at the frontier of research for particle physics. The standard model involves only the three other forces , not the gravitational. A future standard model should have both the present standard model and quantization of gravity, a Theory Of Everything (TOE).
So photons are the building blocks of light, and gravitons are (hopefully) the building blocks of gravitational waves.

*LIGO has just (Feb 11,2016)  announced the observation of gravitational waves https://dcc.ligo.org/public/0122/P150914/014/LIGO-P150914%3ADetection_of_GW150914.pdf
