what does Cold Warrior mean？

The "cold war" was the state of continuous tension that existed between the communist nations led by the Soviet Union and the western democracies led by the US. "Cold warriors" are people who took leadership positions in that struggle.

They (and others) were presidents during the Cold War

http://en.wikipedia.org/wiki/Cold_War

The Cold War (Russian: Холо́дная война́, Kholodnaya voyna, 1947–1991) was the continuing state of political conflict, military tension, proxy wars, propaganda warfare, and economic competition existing after World War II (1939–1945), primarily between the Soviet Union and its satellite states, and the powers of the Western world, particularly the United States.