I would say that the major power that was meant to reside with Congress, but has been taken by the Presidency, is the power to declare war.
Although the president is the commander-in-chief of the armed forces, the Constitution says that it is the role of the Congress to decide when America will go to war. This is, for the most part, no longer how things are done. Presidents have claimed the power to send troops to areas where they will engage in combat without having Congress declare war first. In fact, Congress has not declared war since 1941 and yet the US has fought many wars since then.
By taking the power to commit troops to combat, the Presidency has taken a power that was supposed to be in the hands of Congress.