phoog is right that actually getting rid of the Electoral College would require a Constitutional amendment.
However, it is possible to effectively institute a national popular vote for president without getting rid of the Electoral College, and without a Constitutional amendment, via the National Popular Vote Interstate Compact. Basically, it's an agreement between states to assign their electors to be pledged to the candidate that won the national popular vote. It only takes effect when states with a majority of electoral votes have joined the compact (currently, they have 165 electoral votes, need 270), so that them voting a block guarantees the outcome, regardless of how states that have not joined the compact vote.
There are some complications with this system, e.g. how recounts work if the national popular vote is close (theoretically there should be a nationwide recount, but the compact cannot bind states that have not joined it), and what authority would decide the result if states within the compact disagree about the national popular vote winner. But in general it should be a workable system.