John Adams, our nation's first Vice President, was said to have declared the office the most insignificant that man's imagination had ever conceived. This is the sum total of the constitutional duties required of the Vice President:
(1) To break ties in the Senate. (Article I, Section 3)
(2) To preside over the Senate when it counts Electoral College votes. (Twelfth Amendment)
(3) To discharge the duties of the Office of President as the Acting President if neither the Electoral College nor the House of Representatives is able to elect a President. (Twelfth Amendment)
(4) To succeed the President if the President dies. (Twentieth Amendment, Twenty-Fifth Amendment)
(5) To succeed the President if the President resigns. (Twenty-Fifth Amendment)
(6) To discharge the duties of the Office of President as the Acting President if the President declares himself unable to discharge the duties of the office. (Twenty-Fifth Amendment)
(7) To discharge the duties of the Office of President as the Acting President if the Vice President and the majority of the Cabinet declare the President unable to discharge the duties of the office. (Twenty-Fifth Amendment)
(2) is a duty that has to be fulfilled once every four years. (3) involves being the back-up plan to the back-up plan to the Electoral College; it requires such an unlikely deadlock in our election process that it will be astonishing if it is ever required. (4)-(7) only come into effect if the President dies, resigns, or is declared incapacitated. And the record holder for (1) is John Adams who in eight years broke a total of 29 ties; so even in an extraordinarily deadlocked Senate we would not expect more than three to four cases a year (and almost certainly much less).
The Vice President has no constitutional day-to-day duties. The President is not required to brief the Vice President on anything. The President is not required to allow the Vice President to sit in on Cabinet meetings (and there have been administrations when they were not invited to such meetings). The Vice President, as Vice President, has no official authority to do anything not strictly required by any of the above. The expectation in the Senate is that the Vice President, as President of the Senate, plays a purely procedural role; this is entirely a matter of what Senate rules allow the Vice President to do. The President, of course, can choose to make use of the Vice President as an agent of the Office of the President; but this is entirely at the discretion of the President.
So it's a curious office. It serves three structural functions:
(1) To allow the Senate to have an official presiding officer without requiring any Senator to fulfill that office.
(2) To assist in giving procedural legitimacy to the counting of Electoral College votes.
(3) To prevent the executive branch from shutting down in the case of a loss of the President (for any reason).
In essence, the Office of the Vice President serves as a sort of basket for a handful of our Constitution's miscellaneous back-up systems. As far as constitutional authority goes, it carries with it only two very limited active powers:
(1) To break ties in the Senate;
(2) To declare, with a majority of the Cabinet, that the President is unable to discharge the functions of the Office of the President.
Everything else depends on what other people decide to let the Vice President do. Strictly speaking, the Vice President is not required to do anything on a day-to-day basis: all the required functions of the office are sporadic and rare. Strictly speaking, no one is required to give the Vice President anything to do, ever. It is constitutionally possible to have a Vice President who is not allowed or able to do anything important his or her entire term except open the envelopes that hold Electoral College votes at the end of the term. Of course, some Vice Presidents have been able to turn this into a productive advantage. Apparently, when Thomas Jefferson was elected Vice President, he had nothing to do -- the notion that the Vice President was supposed to be assistant to the President hadn't really taken off yet. So he spent four years working on adapting parliamentary procedure to the needs of the U.S. Congress and working on his campaign for President. Henry Wilson, Grant's second VP, spent much of his time doing historical research, out of which came his three-volume work,History of the Rise and Fall of the Slave Power in America (which you can see online at Internet Archive).