Then you have a democracy, at least in the sense that you have an elected government and all Americans have the vote. Though, as an outsider, your range of candidates does always seem limited to either a Republican or a Democrat.
Technically the electoral college elects the president of the USA- I think that is what fizzle is talking about. Remember the 2000 election? Check out the info on the electoral college: http://en.wikipedia.org/wiki/Electoral_College_(United_States)
But aren't the electors themselves voted for by the public? So the vote counts that way, the same way it would if you voted for individual presidential candidates. (Sorry, I don't remember the 2000 election :/)
Just looked into that 2000 election and read through that Wiki page (very interesting stuff, I learnt a lot). All I can really say is that, while it has its flaws, the United Staes of America does have a functioning democracy. There just need to be some changes and reforms - like the electoral college.
Women's suffrage movement enabled women to vote in states prior to 1920. Kansas, Wyoming and Utah had it in the 1860s. Washington attempted it as early as 1854 but it lost by one vote. The history of this is kind of all over the place. I suppose working accurate history in would kill the joke though .... Fun fact: Although it had been put in place in 1920 Mississippi was the last state to ratify the Nineteenth Amendment (allowing women the vote). They did this in 1984.
It wasn't a true democracy as only some members of society had the vote; it only became a true democracy once women - and therefore everyone - was entitled to elect a leader.
http://www.archives.gov/historical-docs/document.html?doc=13&title.raw=19th+Amendment+to+the+U.S.+Constitution:+Women's+Right+to+Vote