How does light change directions? To the best of my knowledge, light will always travel at the speed of light. How then does it get reflected and change directions? Wouldn't it have to decelerate, come to a stop, and then accelerate in the opposite direction? 
Or would the direction of motion be discontinuous when graphed as a function of time?
 A: Consider reflection from a piece of glass; the glass is made up of amorphous silica, an atom of silicon and two atoms of oxygen.  Light can be modeled as an electromagnetic wave, with the frequency of the light being the frequency of the electric field oscillations.  
The electric field interacts with the material as it penetrates, causing the bound electrons to oscillate in response.  A detailed analysis shows that the incoming light drives the parametric process which generates the outgoing reflected and refracted beams, which will have the same frequency. 
Thus the incoming light does not decelerate; instead it loses energy to the processes which in turn generate the new light which is traveling in the refracted and reflected directions.
Text books such as Griffith's Introduction to Electrodynamics and Hecht's Optics derive these from the actions of Maxwell's equations in dielectric media.  A somewhat different analysis holds for reflections from a metallic surface, but once again the original beam of light is extinguished.  A quantum mechanical analysis is more detailed, but yields essentially the same results while explaining additional details.
A: At a classical level light can interact with electric charge distributions as described by Maxwell's equations. This can be used to explain lens optics which relies on light travelling slower than $c$ in a dielectric medium. To reconcile this apparent contradiction that light should travel slower than $c$, a microscopic picture is needed. I refer you to this stackexchange post for more details on why the speed of light is not $c$ in a dielectric medium.
