In particle physics, helicity is a measure of the component of the spin of a particle in the direction of the momentum. If the spin points along the direction of momentum, the helicity is positive. If the spin points in the opposite direction from the momentum then the helicity is negative. People use different specific definitions of helicity. The definition given in Peskin and Schroeder is
h = (p ⋅ S)/|p|
Most other definitions are essentially equivalent except that they may divide out the magnitude of one, both, or neither of p and S.
I believe the term itself comes from the helix. A helix may come in one of two distinct forms that have different chirality. Imagine putting your thumb along the axis of the helix. As you progress in the direction of your thumb, the helix wraps around either like the fingers of your right hand do when you close your hand, making it "right-handed", or like the fingers on your left hand, making it "left-handed". The two types are different, and no matter how you rotate one it cannot be made to look like the other. Helicity defines a "handedness" to the particle in the following way: Spin is the intrinsic angular momentum of a particle, so one can just visualize a spinning ball moving in some direction. Just like with the helix if you stick your thumb in the direction of momentum the ball is spinning in either the right-handed sense or the left-handed sense.
Helicity comes up in particle physics when studying spin 1/2 fermions like electrons and neutrinos. One can solve the Dirac equation in a representation in terms of left-handed (negative) and right-handed (positive) helicity states. Helicity is most important, however, for the reason that the weak nuclear force interacts differently based upon the helicity of the particles, leading to parity violation. One of the results of which is that all neutrinos seem to be right-handed while all anitneutrinos are left-handed.
Helicity can obviously be generalized to describe a similar relationship in anything with angular momentum and linear momentum. The term "helicity" is also used in some cases to describe a similar sort of relationship in classical fields (like fluid flows or classical electromagnetism), where for a field F a helicity is often defined as
F ⋅ ∇ x F
So, again, it measures the alignment of a rotational component with a linear one.
It's important to note that for massive particles helicity is not a frame independent quantity, because we can always change to a frame where momentum is reversed, but spin will not be reversed. For a massless particle no such transformation is possible and helicity will be the same in all frames. Helicity is a physical quantity associated with a particle in a certain state that can be used to assign a "handedness" or chirality to the particle, but it can only be an intrinsic property of the particle if it is massless.
Sources:
- An Introduction to Quantum Field Theory, Peskin and Schroeder
- http://www.nso.edu/sunspot/ppages/apevtsov/www/helicity_def.html
- Classes I've had, lectures I've attended, etc.
I wrote this for background for another node. My familiarity with this subject is limited, so probably a lot more can be said. | 
