Democracy Demands One Person, One Vote
Some people think that the current system makes votes in small states more important. Actually, it makes voters in closely divided “battleground” states more important. This year’s battleground states will likely include Ohio, Florida, Pennsylvania, Michigan, and perhaps Virginia and Colorado—all larger states. Candidates will pay attention to these states and ignore the vast majority of states (both large and small) because those states are deemed safely “red” or “blue” where it does little good for candidates to try to pick up additional votes.
What’s more, as Americans have increasingly moved to communities of like-minded people, the number of states that are battlegrounds has declined dramatically. In 1960, there were 24 battleground states, but this number had dropped to just 13 states in the 2004 election.
While the battleground states may change over time, we know for sure that most of us will be left on the sidelines in any given presidential election. That’s not what democracy should be about.
A national popular vote would make every vote equal. If we want our government to enjoy the legitimacy and respect that can only come from the consent of the governed, we must elect our president the same way we elect every other officeholder in our country: every vote should count the same and the candidate with the most votes should win.

Though Democracy may say that we should be ruled this way, our Government is not this. We are a Democratic Republic, thus we use representatives. What you are suggesting is a Direct Democracy, other wise known as Mob Rule.
National Popular Vote has nothing to do with whether the country has a "republican" form of government or is a "democracy."
A "republican" form of government means that the voters do not make laws themselves but, instead, delegate the job to periodically elected officials (Congressmen, Senators, and the President). The United States has a "republican" form of government regardless of whether popular votes for presidential electors are tallied at the state-level (as has been the case in 48 states) or at district-level (as has been the case in Maine and Nebraska) or at 50-state-level (as under the National Popular Vote bill).
If a "republican" form of government means that the presidential electors exercise independent judgment (like the College of Cardinals that elects the Pope), we have had a "democratic" method of electing presidential electors since 1796 (the first contested presidential election). Ever since 1796, presidential candidates have been nominated by a central authority (originally congressional caucuses, and now party conventions) and electors are reliable rubberstamps for the voters of the district or state that elected them.
In 1789, in the nation's first election, the people had no vote for President in most states, Only men who owned a substantial amount of property could vote. Later, state laws gave the people the right to vote for President in all 50 states.
The National Popular Vote bill would end the disproportionate attention and influence of the "mob" in a handful of closely divided battleground states, such as Florida, while the "mobs" of the vast majority of states are ignored. 98% of the 2008 campaign events involving a presidential or vice-presidential candidate occurred in just 15 closely divided "battleground" states. Over half (57%) of the events were in just four states (Ohio, Florida, Pennsylvania and Virginia). Similarly, 98% of ad spending took place in these 15 "battleground" states.
The current system does not provide some kind of check on the "mobs." There have been 22,000 electoral votes cast since presidential elections became competitive (in 1796), and only 10 have been cast for someone other than the candidate nominated by the elector's own political party. The electors are dedicated party activists of the winning party who meet briefly in mid-December to cast their totally predictable votes in accordance with their pre-announced pledges.