There is only one President of the United States at a time. While the election for President occurs in November, the President is not sworn in until January. The result is a three month period of time as the current President finishes out his term and the new one prepares to transition into power.
This period of time used to be longer. Up until 1937, the President did not take his oath until March. This five month period often caused difficulties. FDR did not work at all with Hoover and five months were wasted which could have been used to deal with the Great Depression.
The same is true with the Buchanan-Lincoln exchange of power. Wikipedia notes, "The President often works closely with the President-elect on important policy matters during the last three months of the President's term to ensure a smooth transition and continuity of operations that have significant national interests. The historical failure of James Buchanan to do this when he was President and Abraham Lincoln was the President-elect is considered one of the reasons leading up to the American Civil War." I think the Civil War was going to happen regardless. However, had Buchanan worked with Lincoln perhaps the Union would have been in better shape and the war would have ended sooner. However, partisan politics were more bitter in the 19th Century and Buchanan may have had no desire for Lincoln to succeed.
Things have changed a great deal. Bush and Obama have already meet to discuss transition issues. The Federal government now actively helps the new incoming administration as well. Obama and his team have an official government site at Change.gov. You can read news of the transition and even apply for jobs. Check out the Plum Book at http://www.gpoaccess.gov/plumbook/ if you want to try for one of these political appointments. (However, if you had not supported the Obama '08 campaign or have significant Democratic party affiliation, don't hold your breath.)
The President-elect of the United States is not the President. However, he or she will be in a few months after a presidential election when the incumbent is leaving office. It is good to see that reflected by the current administration and the federal bureaucracy.