There are all sorts of arguments for and against keeping the Electoral College.
Perhaps the biggest argument for the Electoral College is that it makes the states, and particularly small states, more important in the presidential election. In this view, an election by popular vote would lead to small states simply being ignored. The candidates would concentrate on the big states with the most voters and they would ignore the other states. This makes it more representative of the country as a whole than a popular vote would be. There is also the argument that having a nation-wide vote would be horrible in the case of a close election. When George W. Bush won the Electoral College but lost the popular vote in 2000, only Florida had to recount their ballots. The margin for the country as a whole was about 500,000 votes (out of 50 million or so) in favor of Al Gore. The chaos that would have ensued if the entire country had had to recount the votes due to the margin being that slim is unimaginable.
As to the arguments against the Electoral College, there are two major points. First, the system makes the votes of many people worthless and it gives them an incentive not to vote. A Democrat in Idaho or a Republican in California has no real incentive to vote in the presidential election. They know there is no way their candidate will win their state. A nationwide popular vote would give them more incentive to vote. Second, the system makes a few “battleground” states more important than all the other states. Only a few states are really likely to be closely contested and so those states become the focus of essentially the whole campaign. This makes the rest of us seem fairly unimportant.