While ancient Athens was an early and important example of a democracy, it was not a true democracy by modern standards. In a true democracy, everyone in the society has a say in how they are governed. In ancient Athens, only adult males were allowed to participate in the democratic process. Women, slaves, and foreigners were excluded, meaning that fewer than a third of adult Athenians had a say in their governance.
Unlike the United States, Athens was small enough that all the qualified citizens could assemble and vote in person on legislative and executive issues. In other words, the government operated like a referendum. In a referendum, which sometimes occurs in the U.S., the public votes directly on a particular issue. Otherwise, in the United States and other modern democracies, representatives are elected to vote on behalf of a geographic group of people.
Athenian democracy marked a departure from monarchial or aristocratic rule, in which a single leader or small group makes all the decisions and hands them down. Democracy was instituted in Athens because the rulers passed laws that benefitted only themselves, while placing burdens and hardships on the rest of the society. Democracy, which is a "bottoms up" system that takes into account people with less power and privilege, helps ensure that everyone has a say in governance, which ideally leads to fairness.