Herbert Hoover (term: 1929-1933) had one of the most unfortunate starts to a Presidency when the Great Depression hit the same year he was inaugurated. The state of the economy can determine a President’s ability to be reelected at times (while it is never that simple, it can be a massive factor in measuring a President’s success). The Great Depression not only crushed the US economy, but also the global economy and led to Franklin Roosevelt exerting serious Presidential power when he unleashed the New Deal.

Dwight Eisenhower (term: 1953-1961) officially saw Hawaii became the 50th state on August 21, 1959. This led to the modern American flag we know now. Hawaii was annexed by America in 1898 in part because of geostrategy - the base at Pearl Harbor provided several advantages during the Spanish-American War.

John Kennedy (term: 1961-1963) was in office during the Cuban Missile Crisis. America and the Soviet Union were bitter enemies during the Cold War and each found allies throughout the world. The Soviet Union deployed missiles to Cuba, a Communist ally, after America had failed to topple the Castro government during the Bay of Pigs invasion. This led to the most direct conflict of the Cold War, and the possibility of war was very very real.

Ultimately, negotiations settled the situation peacefully.

Jimmy Carter (term: 1977-1981) oversaw the Iran Hostage Crisis, which occurred in 1979 and lasted over 400 days. It was a massively public situation that had a ton of angles to it, and many argue that this cost Jimmy Carter a second term in office. 

Iran was facing a revolution, and the Iranian students that held the Americans hostage in the American embassy were supported by the powers that were not backed by the American government.

President Ronald Reagan (term: 1981-1989) sealed an executive agreement with USSR general secretary Mikhail Gorbachev that removed certain close-range nuclear weaponry from government's weapons caches.  

Executive agreements do not need Senate approval, but can lay the groundwork for official treaties and other diplomatic progress.  

There are 538 electors, which correspond to the 100 members of the Senate, 435 members of the House of Representatives, and 3 additional members for the District of Columbia. In order to carry the majority of the electoral votes, a President must win at least 270 electoral votes.

A line-item veto allows the President to block parts of a provisional bill in the Legislature, without having to block the whole of the bill. Although the line-item veto has been proposed numerous times in Congress, it has never been approved by both Congress and the Supreme Court; (the Court ruled it unconstitutional in 1998). Presumably, the Legislative and Judicial Branches fear extending too much power to the Executive to control the exact direction of legislative action.

In the event that the electoral college fails to elect a President then the responsibility for choosing a President falls to the House of Representatives. This happened in the election of 1800, when flaws in the original Constitution saw all members of the Democratic-Republican Party casting their votes for Thomas Jefferson and Aaron Burr, thus creating an equal vote for President and Vice-President for both men. This was rectified in 1804 with the passage of the Twelfth Amendment, which stipulated that the electoral college must clearly cast their votes individually for President and Vice-President. It also happened in the election of 1824, when John Quincy Adams was chosen by the House (and Henry Clay) after receiving less of the popular vote and the electoral vote than Andrew Jackson—the so-called "Corrupt Bargain."

An Executive Order allows the President to pass a law outside of Congress. It allows the President to bypass Congress in making laws. It was originally uncommon in American political behavior, but it has gained prominence since the Presidency of Ulysses S. Grant. The usage of Executive Order peaked with Franklin D. Roosevelt and served as a function of his massive overhaul of social security and welfare in America during the New Deal program.

A line-item veto exists in many countries as means for the Executive to veto a small portion of bill without vetoing the entire package; however, in the United States it has consistently either not been supported by Congress or has been ruled unconstitutional by the courts. The President in the United States cannot use a line-item veto.