There have only been three (or possibly four) such presidents.
The certain ones are Rutherford B Hayes (1876), Benjamin Harrison (1888) and George W. Bush (2000).
The probable case is John Quincy Adams (1824), who was behind Andrew Jackson in the popular vote, but only in those states which used the popular vote to determine presidential electors. At that time several states, including important ones such as New York, did not do so, but allowed the Legislature to choose Electors. There is no way to tell how a nationwide popular vote would have gone.
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Presidents were never elected by popular vote. They have always be elected by the electors from each state who cast their electoral votes. What has changed is the way that the electors are selected. Nowadays they are chosen by popular vote, but at first the state legislatures would choose them.
No US president was elected unanimously by popular vote. The only president elected unanimously by the electoral college was George Washington (There was no popular vote in this election).
Electors are elected by popular vote but the president is elected by the electoral college. A president candidate can win the popular vote and still not win if he doesn't win the electoral college.
All state officials are elected by popular vote.
No. The popular vote for each state determines the ELECTORAL COLLEGE votes, which add up to declare the winner. In this way, if a candidate wins the top states, but not the popular vote, he/she will still become president.