The appointment of a Fed chairman is different than presidential appointments of judges and Cabinet secretaries. The Constitution provides that judges "shall hold their Offices during good Behaviour", which, in practice, means that they stay in office until they resign, die, or are removed via the impeachment process. And Cabinet secretaries can be fired by the president at any time; a quaint phrase has it that they serve at the pleasure of the president.
But the Fed chairman is appointed by the president for a fixed four-year term, subject to Senate confirmation. The incumbent, Ben Bernanke, was originally nominated for the job in 2005 by George W. Bush. Last year, President Obama decided to nominate Bernanke for a second term. That nomination is currently on the Senate's agenda.
For the most part, when the White House changes partisan hands, the new president replaces his predecessor's appointees with nominees from the new president's party. There are exceptions, such as Defense Secretary Robert Gates, who was originally appointed by Bush, and was kept in that job by Obama. But those exceptions are rare.
The Fed job has, however, not traditionally been subject to such partisan considerations. Bernanke's predecessor, Alan Greenspan, was originally nominated by Republican Ronald Reagan, in 1987. Greenspan's appointment was subsequently renewed four times, twice by Democrat Bill Clinton, and once by each of the Republican Presidents Bush. Similarly, Greenspan's predecessor, Paul Volcker, served two terms, originally nominated by Democrat Jimmy Carter, and re-nominated by Reagan.
Bernanke is under attack from both left and right. There is a school of thought that would say that indicates he's finding the correct middle ground. They might be right; I have no firm opinion on whether Bernanke should stay in his job.
The Fed has been subjected to populist attacks from elements across the political spectrum, since it was created in 1913. It has been involved to an unprecedented degree in the bailouts of financial institutions during the recent financial crisis. That has intensified the criticism which, rightly or wrongly, has fallen on Bernanke. And the populist element in the Scott Brown candidacy (can't quite picture Bernanke driving a truck around Massachusetts) might be making the Senate more sensitive to such concerns.
Sewell Chan describes, in The New York Times (read it now, before you have to start paying for access), the current state of play in the Senate.
One of Bernanke's opponents on the left, Senator Bernie Sanders, Independent of Vermont, put this Huffington Post item on his website. Robert Kuttner wrote, on January 11, when it was only starting to become apparent that Brown might win the Senate election, that:
With a majority of Republican senators apparently ready to vote against Bernanke, Democratic senators risk finding themselves on the wrong side of another populist backlash. If half of the Democrats decide to vote against him, his nomination could go down.
Putting it in terms of economist-speak with which Bernanke would be familiar, Washington has been thrown into a state of disequilibrium and, until the politicians find their footing again, anything could happen.