Your eye is more sensitive to green than, say, red.
Also as the laser beam travels, dust and other particles are in the way. This causes a scattering of the beam, allowing you to see it from behind or from the side or indeed from most any angle.
How is it possible to see the light path of a green laser pointer?
Usually reflection off of matter (dust, smoke, water vapor etc), but can be Raleigh backscattering (off of air molecules) if the laser intensity is high enough.
I'm guessing your question is aimed (pardon the pun) at the latter effect. Green lasers will have a light path that is more visible than a red light in clear air for several reasons.
1. The laser power is often higher. (I'm thinking of those high power Argon-ion lasers in light shows)
2. Eyes are >10 times more sensitive to green light than red light.
3. Raleigh backscattering has a 4th power dependence on wavelength resulting in green light reflecting much better than red light by about a factor of 3.