Will the light beam appear to move slower than the speed of light or not ? If not why ? Please tell I don't understand.
It will still move at the speed of light by the second postulate of special relativity.
If the observer moves with speed v (relative to the light source), he will measure the speed of the light to be c'=c-v. Einstein's "speed of light is the same for all observers" (c'=c) was "borrowed" from the Lorentz equations but even the authors of the equations did not believe in it:
https://en.wikipedia.org/wiki/Lorentz_ether_theory Albert Einstein: "...it is impossible to base a theory of the transformation laws of space and time on the principle of relativity alone. As we know, this is connected with the relativity of the concepts of "simultaneity" and "shape of moving bodies." To fill this gap, I introduced the principle of the constancy of the velocity of light, which I borrowed from H. A. Lorentz's theory of the stationary luminiferous ether..."
http://www.jstor.org/stable/3653092 "It is clear from the context that Poincaré meant here to apply the postulate [of constancy of the speed of light] only in an ether-bound frame, in which case he could indeed state that it had been "accepted by everybody." In 1900 and in later writings he defined the apparent time of a moving observer in such a way that the velocity of light measured by this observer would be the same as if he were at rest (with respect to the ether). This does not mean, however, that he meant the postulate to apply in any inertial frame. From his point of view, the true velocity of light in a moving frame was not a constant but was given by the Galilean law of addition of velocities."