The helicity of a particle is right-handed if the direction of its spin is the same as the direction of its motion. It is left-handed if the directions of spin and motion are opposite. By convention for rotation, a standard clock, tossed with its face directed forwards, has left-handed helicity. Mathematically, helicity is the sign of the projection of the spin vector onto the momentum vector: left is negative, right is positive.
The chirality of a particle is more abstract. It is determined by whether the particle transforms in a right or left-handed representation of the Poincaré group. (However, some representations, such as Dirac spinors, have both right and left-handed components. In cases like this, we can define projection operators that project out either the right or left hand components and discuss the right and left-handed portions of the representation.)
For massless particles—such as the photon, the gluon, and the (hypothetical) graviton—chirality is the same as helicity; a given massless particle appears to spin in the same direction along its axis of motion regardless of point of view of the observer.
For massive particles—such as electrons, quarks, and neutrinos—chirality and helicity must be distinguished. In the case of these particles, it is possible for an observer to change to a reference frame that overtakes the spinning particle, in which case the particle will then appear to move backwards, and its helicity (which may be thought of as ‘apparent chirality’) will be reversed.
A massless particle moves with the speed of light, so a real observer (who must always travel at less than the speed of light) cannot be in any reference frame where the particle appears to reverse its relative direction, meaning that all real observers see the same chirality. Because of this, the direction of spin of massless particles is not affected by a Lorentz boost (change of viewpoint) in the direction of motion of the particle, and the sign of the projection (helicity) is fixed for all reference frames: the helicity is a relativistic invariant.
With the discovery of neutrino oscillation, which implies that neutrinos have mass, the only observed massless particle is the photon. The gluon also is expected to be massless, although the assumption that it is massless has not been well tested. Hence, these are the only two particles now known for which helicity could be identical to chirality, and only one that has been confirmed by measurement. All other observed particles have mass and thus may have different helicities in different reference frames. It is still possible that as-yet unobserved particles, like the graviton, might be massless, and hence have invariant helicity like the photon.