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The four president who were elected without popular winning votes

The presidents took office without winning the popular vote.

 In other words, they did not receive a plurality in terms of the popular vote. 

They were elected, instead, by the electoral college or in the case of John Quincy Adams by the House of Representatives after a tie in the electoral votes. They were:

John Quincy Adams who lost by 44,804 votes to Andrew Jackson in 1824
Rutherford B. Hayes who lost by 264,292 votes to Samuel J. Tilden in 1876

Four presidents took office without winning the popular vote. 
In other words, they did not receive a plurality in terms of the popular vote. 
They were elected, instead, by the electoral college or in the case of John Quincy Adams by the House of Representatives after a tie in the electoral votes. 

They were:John Quincy Adams who lost by 44,804 votes to Andrew Jackson in 1824
Rutherford B. Hayes who lost by 264,292 votes to Samuel J. Tilden in 1876
The Twenty-Third Amendment gave the District of Columbia three electoral votes since they have no Senators and Representatives. 

Since states vary in population and many popular votes for different candidates can be quite close within an individual state, it makes sense that a candidate might win the popular vote across the entire United States but not win in the electoral college. 
As a specific example le's say the electoral college is only made up of two states: Texas and Florida. Texas with its 38 votes goes entirely to a Republican candidate but the popular vote was very close and the Democratic candidate was behind by a very small margin of only 10,000 votes.
 In the same year, Florida with its 29 votes goes entirely to the Democratic candidate, yet the margin for the Democratic win was much larger with the popular vote win by over 1,000,000 votes .
This could result in a Republican win at the electoral college even though when the votes between the two states are counted, the Democrats won the popular vote. 

Despite the example given above, it is very rare for a president to win the popular vote yet lose the election. 

As we have stated, this has only happened four times in US History, and only once in the last 100 years. 
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