If you say the US is a democracy, a whole bunch of people will come out of the woodwork to explain that it's a republic and the President is indirectly elected. Certainly the President is not elected by popular vote, a very important fact recently. Nor does the Federal government run the presidential elections.
The US is a republic rather than a democracy because people don't vote directly on laws. Its unrelated to how the president is elected. An example of a democracy would be the Swiss system.
