The Senate's elected leader, known as the president pro tempore, presides over the Senate in the absence of the Vice President. Typically he will cede this role to junior members who will practice the formal parliamentary procedure of the Senate.
The position is separate from the leaders of each party in the Senate, known as the Majority Leader and Minority Leader, and by custom is a senior member of the Senate.
The President pro tem is technically in charge, but in practice various (usually junior) senators rotate through as presiding officer so they become acquainted with the rules of procedure.
the vp of the senate
The Senate is where the VP presides.
The Vice President of the United States is the presiding officer of the Senate and is in charge of it. However, the day-to-day operations of the Senate are overseen by the Senate Majority Leader and the Senate Minority Leader, who are elected by the members of their respective parties.
The Senate President Pro Tempore
The President of the Senate (VP of US) votes to break ties.
The VP of the US chairs the Senate.
The VP chairs the Senate and can vote if a tie-vote occurs. Therefore the party that the VP belongs to can pass legislation with only 50 votes.
He is the VP and not a voting member of the Senate, but he can vote in the event of a tie.
The VP is not a Senator. His job is to chair meetings. He can not vote unless there is a tie vote.
VP Joe Biden
majority leader
There is no Vice-president in that case. That office remains vacant and the duties of the Vice-president to preside over the Senate is then handled by the President pro tempore of the Senate.