Yesterday, President Barack Obama announced his plans to run for reelection in 2012, 582 days before Election Day and before most major Republican opponents officially announced that they'd be entering the race. This is the earliest any incumbent president has officially signed up to run again.
President George W. Bush filed for reelection on May 17, 2003. Several Democratic challengers had announced their intention to run months before, for a general election in November 2004.
President Bill Clinton was earlier than Bush, but later than Obama (in relation to the number of days before the general election): he announced he was running again on April 15, 1995. Clinton's announcement came just days after eventual Republican nominee Bob Dole declared his candidacy. (Even the New York Times was shocked two months later when President Clinton began running--gasp!--television campaign ads in June 1995.) The general election was a year-and-a-half later, in November 1996.
President George H. W. Bush was particuarly late to announce his intention to run for in the November 1992 election, waiting until December 1991 to formally name campaign staff and start the race. Bush was the last incumbent to lose his reelection bid.
President Ronald Reagan did not officially become a candidate until October 18, 1983, giving the Gipper (only!) a year to mount his reelection battle (the general election was in November 1984).