If there was a quantum theory of gravity (which there isn't) then it would include a graviton as the elementary particle associated with gravity, in the same way that the photon is associated with the electromagnetic field (as CuriousMind says).
An electromagnetic wave in vacuum is transverse - both the electric and magnetic fields are vectors that are at right angles to the direction of propagation. In Quantum Field Theory this is somehow connected to the fact that the photon has spin 1.
A gravitational wave is a tensor wave - the disturbance is in the metric, which is a tensor. One can loosely say it is transverse for the following reason:
There is a theorem that says that a spherically symmetric body (for example a symmetric collapsing star) does not generate gravitational waves - because all the movements of matter are radial, and this in the same direction as the putative outgoing wave. The tensor nature means the graviton must have spin 2.
In Quantum Electrodynamics (QED) one or more virtual photons are exchanged between two particles that interact electromagnetically. (I suppose this picture strictly only applies in the perturbation theory approach, in which Feynman diagrams are used.)
In Quantum Gravity Theory (if it existed) one or more virtual gravitons would be exchanged between particles interacting gravitationally. The photon and the graviton are both massless. This is related to the fact that both are long range forces, with an inverse square dependence, with no exponential drop-off with distance.
In QED real photons (observable light) are created when charges accelerate. In Quantum Gravity Theory (if it existed) real gravitons are created when mass (or pressure or stress) accelerates.
The particle nature of gravitational radiation would be difficult to observe experimentally, because it is almost impossible to think of processes that would create gravitons of 1 electron-volt of energy or more - the energy that is easy to measure as it corresponds to chemical reactions, the photoelectric effect, and such-like. Hope this helps, @HDE.