The President of the Senate is the head of the Senate and this function is always served by the Vice-President of the United States.
However, the Vice-President is usually committed to other work and can't in practice we present at the Senate.
The person who leads the Senate when the President of the Senate can't is called the President Pro Tempore of the Senate: this is usually the person with the biggest seniority, that is experience in the Senate.
Currently, it is Orrin Hatch, a Republican.