Inauguration Day is set by the Twentieth Amendment to the United States Constitution:
The terms of the President and the Vice President shall end at noon on the 20th day of January, and the terms of Senators and Representatives at noon on the 3d day of January, of the years in which such terms would have ended if this article had not been ratified; and the terms of their successors shall then begin. [See the National Archives Charters of Freedom site.]
Therefore, at 12:00 noon in Washington, DC, on Inauguration Day 2013 (Sunday, January 20, 2013), there will be a ceremony to swear in the newly elected (or re-elected) President of the United States. Either Barack Obama will continue as the 44th President of the United States for four more years, or a Republican challenger will become the 45th President.
However, because the Constitutional date for the transfer of power is a Sunday, the public ceremonies will almost certainly take place on Monday, January 21, 2013 -- particularly if President Obama is re-elected. (His predecessors Ronald Reagan and Dwight Eisenhower were sworn in for a second time in private on Sundays and in public the day after. So far, a first-term president has not taken office on a Sunday. 2013 could be the first time a new president was sworn in on a Sunday since Zachary Taylor on March 4, 1849 (the date of the Inauguration was changed from March 4 to January 20 in 1933).
This site will track the likely nominees to go up against President Obama in the November 2012 election and then the subsequent preparations for the Inauguration several months later.