Arthur Coleman Winters was the President of the United States of America circa Spring 2008. After "Harold Saxon" announced the existence of the Toclafane to the world, Winters served as designated United Nations ambassador to the Toclafane.
After Winters angered Saxon (actually the Master), Saxon ordered Winters killed by one of the Toclafane. His death signaled the start of the Toclafane occupation of Earth. (DW: The Sound of Drums)
The Paradox Machine only rolled back time to a few moments after Winters' death, meaning that it still happened. (DW: Last of the Time Lords)
It is also likely he was the President mentioned in The Christmas Invasion by Harriet Jones.
Behind the Scenes
Upon meeting the Toclafane, Winters refers to himself as "President-Elect", suggesting that he has recently won an election but has not yet been sworn in. However, during The Sound of Drums, he is referred to as "President" numerous times and by different characters (The Doctor, "Saxon", Trinity Wells and others) and once time is reset near the end of Last of the Time Lords, UNIT can be heard over the radio saying they've just seen "the President" killed.
Also, unless American law differs in the Whoniverse, it is impossible for an individual to be called the "President-elect" in the spring as the title is only used by the winner of the US federal election which, by law, is held on Nov. 4, with the new president sworn in on the following Jan. 20, well before the events of The Sound of Drums.
Explanations for this could be the character being nervous and/or confused in front of the Toclafane, or Russell T. Davies not understanding the meaning of the term "president-elect".