What role did the Hitler Youth play in the actual fighting of World War II?
The role of the Hitler Youth in the actual fighting was somewhat limited, but nonetheless was still an important one. While the German military was fully engaged on all fronts by the later years of the war, the Hitler Youth were used to prepare defenses (such as tank traps) to supervise slave laborers and even to administer small towns during such projects. This freed up regular army troops for more important military duties elsewhere.
But members of the Hitler Youth received military and weapons training during most of their time in the organization, so they were easy to convert to regular soldiers (and SS members) once they reached adulthood, as many did during the war years. The Hitler Youth also had a flight training program that fed pilots into the Luftwaffe, which was critical given the losses German flight crews had experienced after 1941.
If you can find it, read A Child of Hitler by Alfons Heck, who was himself a 17-year old Major in the Hitler Youth. Very enlightening look from the inside of the organization, and an explanation of how effective brainwashing can be. He is very regretful for the role he played in the war.