When thinking about political power and influence it is important to distinguish between formal and informal power.
While the law explicitly provides some powers, often times the most significant powers are entirely informal (not defined in law).
The Vice Presidency is a position that largely relies on informal power.
Who Runs the Senate?
According to the Senate website , historically the Vice President's major job was presiding over the Senate.
Their job included administering the rules of the Senate, casting tie-breaking votes, and overseeing Senate floor meetings.
The Vice President's expenses were also funded through legislative appropriation at that time.
This changed in the 1961 when Lyndon B. Johnson moved the Vice President toward an executive-focused role, rather than a legislative one.
Currently the duties of overseeing the Senate are done by the President pro Tempore of the Senate .
However, that position is largely viewed as being ceremonial or administrative rather than important, so it is often delegated to junior Senators to help them learn Senate procedure.
So What Does the Vice President Do Now?
As mentioned in Ryathal's answer (and implied in the question) the Vice President doesn't have much Constitutional authority.
However, they have a large degree of authority vested in them by the President (see this BusinessInsider article for an interesting discussion).
One interesting facet is that because the Vice President is elected, the President can't fire them - unlike other advisors.
Stemming from this position, the Vice President has statutory (but non-Constitutional) roles - such as being a member of the National Security Council .
The Vice President is executive staff, and an advisor without any particular portfolio.
So although they aren't vested power by the Constitution, their status as an executive officer, combined with being elected alongside the President, means that informally they are often have significant influence.
politics.stackexchange.com/users/3169/indigochild
The vice president is almost entirely a ceremonial position, the major purpose is to become the president at a moment's notice.
The president is so important that it's worth it to have a full time back up.
There is one other duty the Vice President has, he opens/presents the ballots from the states for presidential elections.
Being head of the Senate is essentially something for the Vice President to do while they are waiting.
The vice president does have a lot of soft power, traditionally this was focused on legislative lobbying, but more recently the vice president has become part of the leadership team similar to a cabinet member.
politics.stackexchange.com/users/1827/Ryathal
Actually, the Vice President has enormous power.
There are only two people in the USA who are elected by all 50 states (in a "national election") – the President and the Vice President.
This means that the President cannot fire the Vice President.
As both were elected by the people, the VP doesn't work for the President.
Each can be removed only through the impeachment process.
( See Article 2 Section 4 of the US Constitution )
Traditionally, the person occupying this office has been low key and subservient to the President.
But that's done out of loyalty and friendship, not by any constitutional or other legal requirement.
A rogue VP could be disastrous for a sitting President.
The VP commands national media attention, can draw huge crowds and can have a major impact on government, politics, culture, the economy, etc.
In fact, he/she is in a strong position to challenge the President on any matter, including the office itself.
That's power.
The President, of course, also has leverage.
He/
She can at any time isolate the Vice President, taking everything from him/her, except that which the President cannot take away – the job itself and the two roles required by the Constitution: President of the Senate and second in line of succession.
The President can also drop the VP as a running mate during a re-election campaign.
But while the Vice President is in office, that person can have huge influence and
his/her job is as secure as the President's.
So it would be wise for a President to keep the Vice President happy.
politics.stackexchange.com/users/12464/Michael Benjamin
The other "enumerated power" is provided by the 25th Amendment (section 4).
Vice President, in conjunction with a majority of department secretaries, can declare that the President is unable to discharge powers and duties of his office and temporarily suspend him.
politics.stackexchange.com/users/10431/Dmitry Rubanovich
The VP breaks tie votes in the Senate.
The most important power of the Vice President that has not yet been mentioned here is that, in addition to presiding over the Senate, the Vice President is also the TIE-BREAKING VOTE in the Senate.
This power can significantly affect the course of the country when you consider the Senate is responsible for approving Federal Court Judges and Supreme Court Justice nominations by the President.
Which have become quite partisan lately.
U.S. Constitution, Article I, Section 3
The Vice President of the United States shall be President of the Senate, but shall have no Vote, unless they be equally divided.
Source:
U.S. Senate Since 1789, 268 tie-breaking votes have been cast.
Presiding over the Senate and casting the tie-breaking vote are the only powers the Constitution enumerates to the VP.
However, traditionally, the VP also performs whatever tasks and duties the President requests.
As a result, some VPs have wielded significant influence and practical power in the White House.
The most notable recent example being VP Dick Cheney under President George W. Bush.
It's also worth noting that the President has no power to fire the VP as the Constitution requires that the VP serve his/her full term unless impeached.
The POTUS can, however, if he runs again, choose a different running mate for VP.
politics.stackexchange.com/users/5971/Rain Willow
According to the Constitution, the Vice President is the head of the Senate, but doesn't really explain what that means except to say that he doesn't get to vote except to break a tie.  Aside from that and being next in the line of succession, what power does the Vice President actually have?  Can he set the Senate's agenda and force calls to order in the Senate the way the Speaker can in the House?
What power does the Vice President really have?
politics.stackexchange.com
<united-states><congress><senate><senate-rules><vice-president>
politics.stackexchange.com/questions/27577
politics.stackexchange.com/users/9351/Wes Sayeed
The vice president does have a lot of soft power, traditionally this was focused on legislative lobbying, but more recently the vice president has become part of the leadership team similar to a cabinet member. the Vice President, in conjunction with a majority of department secretaries, can declare that the President is unable to discharge powers and duties of his office and temporarily suspend him  The most important power of the Vice President that has not yet been mentioned here is that, in addition to presiding over the Senate, the Vice President is also the TIE-BREAKING VOTE in the Senate. The Vice President of the United States shall be President of the Senate, but shall have no Vote, unless they be equally divided.    The VP has soft power and can can temporarily suspend the President. Furthermore, the VP has the overarching vote in the Senate.
The VP has soft power and can can temporarily suspend the President. The VP has the overarching vote in the Senate.