The fact that Greek city-states had evolved from something like an Absolute Monarchy to any kind of shared governmental power is remarkable, particularly since they were the first culture to do so. The word "democracy" itself means governance by the people. It did not imply that it would be inclusive, but the fact that prominent citizens now had a say in government where before they did not is the key point. Our modern notion of a democracy implies that all people have a say in the government, but that was not the ancient Greek notion.
By our modern standards, ancient Greek democracy was not very democratic, at least not in most ways.
The one thing that was, arguably, more democratic about their system was the fact that all voters were allowed to give their opinions on (and vote on) various issues. This was a "direct democracy" as opposed to our "representative democracy" where we only get to vote for the people who make the laws (rather than on the proposed laws themselves).
However, very few Greeks were allowed to vote. No women were allowed to vote. Only men who had a certain amount of property were generally allowed to/able to hold office. It is estimated that something like only 10% of all the people in Athens were allowed to vote. So in that way it wasn't very democratic.
The previous response articulated quite well the idea of how the principles of pure democracy might not have been in action in Ancient Greece. Another element to incorporate into this schematic is how the Athenian notion of the political good was a homogeneous notion. This helped to create a setting where pure democracy was challenging. There was little in the way of diversity on many levels. Professionally, ethnically, intellectually, as well as geographically (in terms of the city- state) were all levels of singularity. This lack of divergence helped to create a setting where monist notions of the good were present. This helped to create the idea of democracy, but not in the fullest of senses.
Greek city states are one of the earliest democracies. These independent city states which included the cities plus outlaying areas, were governed democratically as early as 400 B.C. Around 300 B.C. Greek philosophers and thinkers like Aristotle and Plato advocated and wrote about rule by law rather than be individual.
Cities like Athens practiced direct democracy in which every males in the city was a permanent member of the assembly, that governed the city state. The assembly was the supreme authority which passed law and decided on all important government policies.
Democracies in these states was not only most democratic in their times but also for many other democracies to follow. Some people have called these democracies not democratic enough for not allowing voting rights to slaves and women. But this will not appear so bad a blot on Greek democracies when we consider that even great modern democracies like USA did not gave voting rights to slave for a long time. Even women were given voting rights in USA only after 1920.