What is the relationship of Athenian to modern United States democracy?
Although both the United States and ancient Athens are described as democracies, the systems of government are quite different.
Since the advent of women's suffrage in the early 20th century, all citizens of the US over 18 who have not committed felonies are able to vote. In Athens, women, slaves, metics (resident aliens), and those who did not have Athenian grandparents could not vote.
Another salient difference is that Athens was a democracy and the United States is not. It is a Republic, modelled on republican Rome. Citizens (accept in certain states, such as California) generally do not vote directly for laws but instead vote for representatives to legislative assemblies such as Congress; those representatives vote for laws. Athens was a direct democracy in which citizens were part of the assembly and voted and spoke for themselves rather than through representatives.
Finally, for many important positions, people in Athens were selected by lot -- a blindfolded priestess from Delphi would reach into a urn containing potshards each with a name -- to avoid corruption, whereas selection by lot is rare in the US (except for limited use in jury selection)
The Athenian democracy is where we in the United States obtained the ideas for our democracy. A democracy is a form of a government where every citizen has a chance to vote on policies. The term citizen in ancient Athens and in the early days of the United States meant land owning men. However, constitutional amendments throughout the history of the US has expanded that to mean everyone 18 years or older. The founding fathers wanted a government for the people and by the people based off of Athens. However, Athens was a small location and it was easier for people to travel into the city when it was their turn to serve in the government. This would have been a major problem for people in the United States - traveling to a central location to participate in government - so a representative form of democracy was created. In this, citizens vote for people to represent them in local, state, and federal governments. This is democratic because at the end of their term (which varies depending on what office you hold), that person can be re-elected if they represented those that voted for them well or they can be replaced by someone else.