A divided Senate with equal numbers of Democrats and Republicans (and Independents who may caucus with them). Any vote that results in a deadlock would be decided by the vote of the Vice President, consistent with Article 1 of the Constitution.
War of the Worlds
The Democrats have enjoyed strong majorities in the Senate during the first six years of the Obama Administration, which will most certainly change after tonight. Republicans are expected to make strong gains in the Senate and to take the majority for the remaining two years of Obama's presidency. If the GOP falls short and only picks up five seats, however, the result would be a 50-50 split, assuming of course that the Independents in the Senate caucus as anticipated. Vice President Biden would thus cast the deciding vote in favor of the Democrats, and they would retain their Senate majority.
A Wrinkle In Time
Only twice in history has there been an equally divided Senate, although the Vice President has cast nearly 250 tie-breaking votes. The first divided Senate occurred in the 1880s and the second in 2001. During the unprecedented 107th Congress, the Senate changed hands multiple times: as sitting Vice President, Al Gore tipped the 50-50 split in favor of the Democrats for the brief period between the swearing in of the new Congress and the inauguration of George W. Bush; in office, Dick Cheney then held the decisive vote, switching control to Republicans; and finally, the change in party affiliation by Vermont Senator Jim Jeffords once again changed the balance back to Democrats. During the Bush Administration, Vice President Cheney cast 8 tie-breaking votes, and Vice President Biden has thus far cast zero.
Adding yet further complication, the outcomes of the contested Senate races in Louisiana and Georgia may not be known until subsequent December and January runoffs. Nevertheless, it is unlikely that Vice President Biden will cast many tie-breaking votes in the 114th Congress due to the changing nature of the body. Add the small margin of control for either party to the increasing use of the filibuster, and bipartisan support will be needed to clear the 60-vote threshold, regardless of the party in power.