Answer:
Under the US Constitution, the Vice President of the United States is the President of the Senate. In modern times, the Vice President does not actually preside over the Senate, except on the most rare of occasions or when his vote may be needed to break a tie.
In the absence of the Vice President, the President Pro Tempore of the Senate (or his/her designee) presides. By tradition, the most senior senator of the majority party is the President Pro Tempore of the Senate. Presently, that is is Sen. Robert Byrd of West Virginia. Sen. Byrd is also the oldest and longest serving senator.