President of the Senate
Article I, Section 3 of the U.S. Constitution states that the Vice-President of the United States shall be the President of the Senate. This is a largely ceremonial role; most of the Senate's business is managed by the Senate Majority Leader and other members of the Senate Leadership. Even the ceremonial duties of the President of the Senate are usually handled by his understudy, the President pro tempore.
The Vice-President does have some real power as President of the Senate. He presides over impeachments of federal officers other than the President of the United States. (Because the Vice-President is next-in-line for the presidency, he has a conflict of interest if the president is impeached; in this case, the Chief Justice of the United States presides.) More commonly, the Vice-President can cast the deciding vote when the Senate is otherwise tied on an issue. Then-Vice-President Al Gore famously used this power to support President Bill Clinton's deficit-reduction bill in 1993. Ten years later, Vice-President Dick Cheney would use the same power to pass President George W. Bush's $350 Billion tax cut.
The President of the Senate's counterpart in House of Representatives is the Speaker of the House, although the Speaker is a not only a true member of the chamber he presides over, he has real management power over the House as well as ceremonial duties corresponding to the President of the Senate's.