If the winner of the popular votes does not win the presidency what happens?

A person legally does not need a single popular vote to win a U.S. Presidential Election. It is up to each state to decide how it determines who gets appointed to the electoral college. For the past 150 years or so, with a few minor exceptions, every state has used the popular vote to make the determination, but if a state were to go back to having the state legislature appoint the electors, while it wouldn't be a popular move, it would be entirely within its rights.

In order to win a presidential election, a person must receive votes for president from more than half of the appointed electors. However, if nobody gets that many votes, the electoral vote results matter only as far as determining from among who the House of Representatives will be electing the President.