Does light have mass? 
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*Does light have mass?

*If yes, will it exert force?

*If no, then how are light particles are travelling at light speed?

*If light doesn't have mass how is it attracted by gravitational force (black holes)?

*And what actually is light? 
 A: Photons having zero rest mass is precisely the reason why they travel at light speed. A particle can have momentum without having mass, the only condition is $$(mc^2)^2=E^2-(pc)^2$$
in the case of light, left side is zero and $E=pc$ where $E$ is energy and $p$ is linear momentum.
Furthermore, the gravitational attraction isn't really attraction in the traditional sense. Gravitation curves the space-time so that the geodesic (the locally straightest and shortest path, which the light takes) is curved. So the light just goes along its way, as straight as it can get, and the curvature of the space makes this geodesic path deflect in angle when the ray passes a massive object.
Light is an electromagnetic wave - change in electric field induces magnetic field, and change in magnetic field induces electric field again... this cycle propagates in space with the speed of light.
A: As orion stated, photons in vacuum have zero rest mass and travel at light speed. There are, however, some cases where you can have massive photons, such as in the Abelian Higgs model which describes $U(1)$ symmetry breaking. This model applies to superconductivity : in a superconductor you have the Meissner effect, which is the fact that magnetic field penetrates over a very small depth inside of the superconductor. Its is a signature that the electromagnetic forces become short-ranged, or, in a field theory language, that the photon becomes massive.
