Tie votes in the Senate are broken by the Vice President of the United States. The Vice President serves as the President of the Senate, though they do not participate in votes unless serving as a tiebreaker. This process is outlined in Article I, section 3 of the Constitution.
Tiebreaker votes are not common, but they are not exceedingly rare either. Since 1979, there have been 244 votes which have resulted in a tie that needed to be broken. John Adams was called on to cast the most tie-breaking votes at 29, between the years of 1789 and 1797. As of October 2014, twelve Vice Presidents, including Dan Quayle and Joe Biden have never had to vote in the Senate.