The NY Times has an "explainer" for the Electoral College. It's not the worst I've seen -- indeed, it is better than a lot that I've seen. But there are problems. The framing of it highlights one of the major ones:
It remains one of the most surprising facts about voting in the United States: While the popular vote elects members of Congress, mayors, governors, state legislators and even more obscure local officials, it does not determine the winner of the presidency, the highest office in the land.
There is nothing surprising about it. Members of Congress, mayors, governors, state legislators, and local officials are all elected by popular vote within a single state. None of them are elected by a national popular vote, and none needs to be, and indeed, it would obviously be stupid for any of them to be. The President cannot be elected this way for the obvious reason that the President doesn't represent a state but the union of states. Thus the President will necessarily always be elected by a completely different kind of election, governed by different rules, and this would still be the case if the President were elected by national popular vote.
This leads to an intense focus on battleground states, as candidates look to boost their electoral advantage by targeting states that can help them reach the needed 270 votes of the 538 up for grabs.
The Electoral College doesn't lead to "an intense focus on battleground states"; all election systems have battleground regions, and in the United States, campaigns naturally structure themselves by states because it's a convenient already-existing unit. What the Electoral College does is create a situation in which the battlegrounds are not just the highest-populated states, but shift around from election to election.
Can a president lose the popular vote but still win the election?
No, because there is no national popular vote at all in the United States, and therefore none to lose. Each state runs a different election under different election laws. If you vote in California, who can vote, how you vote, and how the vote is counted depend on Californian election law, and if you voted in another state, all these would be different. In some states, felons in prison can vote. In others, they can't. Means of voting that are allowed in one state are not allowed in another. If you fill out your ballot incorrectly in most states, you spoil your ballot; in another state, like California, they will attempt to contact you in order to see if they can get a correct ballot in time. There are broad similarities, for the obvious reason that they are the same general kind of election, but votes in one state cannot be added to votes from another state without violating the unit rule of addition. Add the votes of a state like California, which is not very strict about voter identification, not very strict about how you have to fill out your ballots, very generous in attempting to make sure that your vote will be counted if you tried to vote (even if you bungled it), to a state which is none of these things, and you aren't adding equal things and the number you get does not, on its own, mean anything at all. You might as well try to add votes in Delaware to votes in Japan and pretend that the number gives you a precise measure of support for something. Votes are legal constructs: they just are what the laws make them in a jurisdiction, and a different election-law jurisdiction has completely different votes -- votes which cannot coherently be added together across jurisdictional lines.
What happens in the Electoral College is that we run not one but fifty-one distinct Presidential elections (fifty states plus the District of Columbia). Each of those elections can be seen as a simulation that answers the question, "Who would be President if the whole country were like this state?" We then partly weight those according to population in the same way that Congressional representation is partly weighted according to population. Thus the Electoral College votes. The apportionment of these votes is going to have to be done either by the state government agencies, as automatically following from their official vote tally, or by another body that has the room to assess whether the official vote tally accurately gives the sense of the state. As it happens, we use the latter, although attempts to restrict the room for assessment have been common over the history of the institution. Congress then counts the vote, so that the election of the President is not in the hands of any one government power: it requires the state governments, independent bodies created particularly for the purpose, and Congress.
There are arguments that the states with smaller populations are overrepresented in the Electoral College, because every state gets at least 3 electors regardless of population. In a stark example, sparsely populated Wyoming has three votes and a population of about 580,000, giving its individual voters far more clout in the election than their millions of counterparts in densely populated states like Florida, California and New York.
Again, the voters in Wyoming are voting in the Wyoming presidential election, which counts for less than the California presidential elections; voters in California are voting in the California presidential election, which counts for more than the Wyoming presidential election. This is, of course, generally true; voters in sparsely populated states are voting in elections that count for less, voters in more heavily populated states are voting in elections that count for more. There is no way to compare them as to 'clout' at the individual level. When political scientists attempt to pin down something like this, they use a very abstract and roundabout way of doing it, like the probability that an individual vote will affect a vote in the Electoral College that will decide the election. Needless to say, or it should be needless to say, this is an entirely artificial way of looking at votes, and in any case necessarily depends in reality on a number of variables that vary over time.
The idea has public support, but faces a partisan divide, since Republicans currently benefit from the electoral clout of less populous, rural states.
This is entirely false; you have only to look at the less populous states to find that they divide about evenly between Republicans and Democrats, with some shifting over time, but not much. Whether a state counts as 'rural' depends entirely on how you are defining 'rural'; perhaps the almost-truth in the above claim is that Republican political rhetoric is more closely tied to supporting the heartland, farmers, miners, etc. Republicans are also under no illusions: The reason why Democrats are so likely to support abolishing the Electoral College is entirely that they think that by doing so they could more easily leverage their control over the dense population centers of New York City, Chicago, and Los Angeles in presidential elections, and, whether or not this is true, Republicans are not going to go along with what would transparently be a Democratic attempt to gain electoral advantage over Republicans.