Why can the popular-vote winner lose?

Because the US presidency is decided by electoral votes, not by the nationwide popular vote. A candidate can win the most votes nationally while losing the Electoral College if their support is distributed inefficiently across states.

How does this happen mathematically?

Under winner-take-all rules, a candidate who wins California by 5 million votes receives the same 54 electoral votes as if they had won by 500,000. Those extra 4.5 million votes are "wasted" in Electoral College terms—they add to the popular vote total but yield no additional electoral votes.

Meanwhile, an opponent who wins multiple states by narrow margins accumulates electoral votes efficiently. If they win Pennsylvania by 50,000 votes, they receive all 19 electoral votes; if they win Michigan by 10,000 votes, they receive all 15. Each narrow victory is worth the same as a landslide.

The 2016 example

In 2016, Hillary Clinton won approximately 2.9 million more votes nationally than Donald Trump. However, Trump won the Electoral College 304–227 (based on the final certified count after faithless electors). Clinton's popular vote margin came largely from overwhelming victories in California (+4.3 million) and New York (+1.7 million), while Trump won Pennsylvania, Michigan and Wisconsin by a combined margin of roughly 77,000 votes—earning 46 electoral votes from those three states alone.

Is this a bug or a feature?

Defenders of the Electoral College argue it ensures presidential candidates must build geographically broad coalitions rather than focusing only on population centres. Critics argue it makes some votes worth more than others and can undermine democratic legitimacy when the popular vote winner loses.

Several reform proposals exist, including the National Popular Vote Interstate Compact (an agreement among states to award their electors to the national popular vote winner) and constitutional amendments to abolish or modify the Electoral College. None has yet achieved the necessary support.