The Electoral College effectively isolates problems and challenges to a state or two -- for example, to Florida in 2000. National popular election of a President would be a disaster if a recount becomes necessary. Furthermore, in a national popular election the candidates for President would focus their attention on the most populous states where the election would always be won or lost. The Electoral College doesn't guarantee that less populous states get attention, but it mitigates the problem. In particular, the Electoral College forces candidates to fight it out in so-called swing states, which might or might not be populous. People who live in California and New York, especially media pundits, don't like that. But I think the alternative of having a mere handful of states choose the President is worse.
Faithless electors who don't vote according to the wishes of the electorate are often identified as a problem in the Electoral College. In North Carolina, this is addressed adequately: by action of law, faithless electors are immediately removed from office and charged with a crime, while their votes are recast by their replacements. If federal law were to read this way, there would be no faithless electors.
Winner-takes-all at the state level is also cited as a weakness of the Electoral College. Actually, it's left to the discretion of each state. Two states do use proportionate representation based on outcomes in congressional districts, but I find that worrisome. As we have seen recently in North Carolina, the party in power after a national census always gerrymanders congressional districts to favor their own party. Last month most North Carolinians voted for a Democratic candidate for Congress, but the resulting congressional delegation from North Carolina is 9-4 Republican. Proportionate representation would add fuel to this fire, and in case you wonder it would not have elected Al Gore in 2000.
It's true that if a third candidate gets enough traction to prevent a majority in the Electoral College, the election will shift to the House of Representatives who will likely ignore the third candidate. But one can argue that the House, although shaped by gerrymandering, is the most direct and contemporary expression of the will of the public electorate in Washington.
I see nothing to be gained by abolishing the Electoral College, although I would like to see federal legislation -- better yet, a constitutional amendment -- to prevent faithless electors.