Since the United States was one of the largest countries in the war, its military contribution was vast.
First, the US contributed by providing military supplies to its Allies. The Lend-Lease Program, for example, gave huge amounts of all sorts of military supplies to both England and the Soviet Union.
Second, the US was responsible for a great deal of the actual fighting. The US did much (but not most) of the fighting in Europe and practically all of the fighting in the Pacific. If you are looking for specific battles the US engaged in that were important you could list such things as the invasion of Italy and the Battle of the Bulge in Europe and the battles of Saipan, Iwo Jima, and Okinawa in the Pacific.