Arthur Coleman Winters was the President of the United States circa spring 2008. After "Harold Saxon" announced the existence of the Toclafane to the world, Winters was said to be furious at Great Britain having taken unilateral action. The night before official first contact, Winters flew to the United Kingdom in the presidential aircraft and met with Saxon. Refusing to acknowledge Britain's authority in the matter, Winters had UNIT seize control and designated himself as the United Nations ambassador to the Toclafane, overriding Saxon.
Aboard the UNIT ship Valiant, which was to serve as the neutral ground meeting point, Winters ordered all armed personnel to withdraw in accordance with UN strictures before the Toclafane appeared. Much to his confusion, they disregarded him and asked for their "Master". Saxon, who in actuality was the renegade Time Lord known as the Master, then stood up and identified himself. Still not understanding the situation, Winters accosted him and demanded an explanation. In response, 'Saxon' stated that he was taking control and ordered one of the Toclafane to kill Winters. One immediately did so, firing a laser bolt that disintegrated the President on the spot. His death, witnessed by the entire world, signalled the start of the Toclafane occupation of Earth. (TV: The Sound of Drums)
When the paradox machine allowing the Toclafane to exist was destroyed, the natural cycle took over and though most of the events that had taken place were reversed, it only rolled back time to a few moments after Winters' death, meaning that it still happened and Winters remained one of the few casualties of the Master's failed plot. (TV: Last of the Time Lords)
Pompous, arrogant, cynical and short-tempered, President Winters saw himself as the rightful Earth representative. He was antagonistic towards the British, declaring that they had elected an "ass" with Saxon as Prime Minister, and displayed what he saw as perfect patriotism, but was in fact textbook arrogance. Winters was also disrespectful and disregarding of the United Nations and the Unified Intelligence Taskforce, perfectly willing to use them to usurp Saxon but otherwise shunting them aside, branding the First Contact as US-led with his own seal, not the UN's. The Toclafane regarded him as stupid. (TV: The Sound of Drums)
Behind the scenes
When meeting the Toclafane, Winters referred to himself as "President-Elect", suggesting that he had recently won an election but had not yet been sworn in. However, during The Sound of Drums, he was referred to as "President" numerous times and by different characters. Moreover, his plane was designated as Air Force One, which would not be true were he merely the President-elect.
Unless American law differs in the Whoniverse, it is impossible for an individual to be called the "President-elect" in the spring. The term applied to the winner of the US presidential election during the brief period from Election Day in November to Inauguration Day on the following 20 January — well before the events of The Sound of Drums. Additionally, a President-elect has no official standing or duties until formally taking office and becoming President, so Winters being on his mission at all would be extraordinarily unlikely were he still only President-elect.
This reference to the "President-elect" was almost certainly a production error that can be jointly laid at the feet of the script editor, the writer and the continuity supervisor, since it only occurs once in the script.