Under what circumstances can the winner of the popular vote fail to win the presidency?

Under what circumstances can the winner of the popular vote fail to win the presidency?
Counting electorial vote 1921, 2/9/21

If you're a United States citizen, 18 years of age or older, you probably think you have the right to vote for presidential candidates in the national election. That's partially correct. When citizens cast their ballots for president in the popular vote, they elect a slate of electors. Electors then cast the votes that decide who becomes president of the United States.

Usually, electoral votes align with the popular vote in an election. But a number of times in our nation's history, the person who took the White House did not receive the most popular votes.

The founders thought that the use of electors would give our country a representative president, while avoiding a corruptible national election. The Records of the Federal Convention of 1787 report that,

...[T]he members of the General Convention...did indulge the hope [that] by apportioning, limiting, and confining the Electors within their respective States, and by the guarded manner of giving and transmitting the ballots of the Electors to the Seat of Government, that intrigue, combination, and corruption, would be effectually shut out, and a free and pure election of the president of the United States made perpetual.

The Debates on the Adoption of the Federal Constitution, explains what might seem like a convoluted system to voters today:

Each state shall appoint, in such manner as its legislature may direct, a number of electors equal to the whole number of senators and members of the House of Representatives to which the state may be entitled in the legislature.

But no person shall be appointed an elector who is a member of the legislature of the United States, or who holds any office of profit or trust under the United States.

The electors shall meet in their respective states, and vote by ballot for two persons, of whom one, at least, shall not be an inhabitant of the same state with themselves.

Under what circumstances can the winner of the popular vote fail to win the presidency?
National political chart & map of the United States.

In 1796, Federalist John Adams was elected the nation's second president, and Thomas Jefferson, of the Republican Party, was elected vice president. On December 28, 1796, Jefferson wrote a letter to Adams, observing, "The public & the papers have been much occupied lately in placing us in a point of opposition to each other. I trust with confidence that less of it has been felt by ourselves personally." How did their political differences actually affect their leadership?

Initially, electors cast votes for candidates without designating whether they were voting for president or vice president. The flaws in this system became evident in 1800 when Thomas Jefferson and Aaron Burr each received 73 electoral votes. It took the House 36 votes before the tie was broken and Jefferson took office as president.

In 1804, 12th Amendment to the Constitution made sure that electors designate their votes for president and vice president, but the 12th Amendment leaves in place a tie breaking system established by the Constitution by which the House of Representatives breaks a tie on presidential electoral votes and the Senate breaks a tie on vice presidential electoral votes.

In some elections, the Electoral College has voted presidents into office by extremely slim margins, as was the case in 1960, when John Kennedy defeated Richard Nixon by fewer than 120,000 popular votes. Electors have even failed to vote for the candidates to whom they were pledged, as was the case when an elector pledged for Michael Dukakis voted instead for vice-presidential candidate Lloyd Bentsen.

Under what circumstances can the winner of the popular vote fail to win the presidency?
United States presidential election 2012, results by state, November 6, 2012

While these electoral methods may seem strange to us now, it may seem even stranger that the founders didn't provide a process by which to nominate presidential candidates. They seemed to expect that candidates would be as obvious and unanimous choices in the future as George Washington had been in their time.

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