Light has relativistic mass
It means that it has energy which can be transformed into mass [E = mc^2]) 
It does not have rest mass
Because light cannot be at rest 
It does not experience a gravitational pull 

It can be affected by gravity though because gravity essentially curves space-time. Since light always goes in a straight line relative to space-time, if it encounters a curvature in space-time it will continue going straight but since space-time is curved, "straight" will be curved as well. In the case of black holes, the curve is so steep that the "straight" path actually spirals down to a single point from the perspective of an outside observer.
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Light has no weight because its just a stream of particles called photons. these have negligible mass so, light has no weight.