This is a very broad question for two reason. First, Greek civilization spans a rather long time. So, there are changes that take place. Second, there are may different city states. So, there are many variations as well. With that said, here are three generalizations.
First, male citizens were free. They had may privileges. They had the right to join the community in all its public activities, the right to vote, the right to the protection of the law, and also the right to stand as jurors. In a place like Athens, a democracy, must was done by drawing lot. Hence, there was not only much freedom, but also much responsibility.
Second, all of these freedoms were to be used for the city state. Hence, there was little idea of individualism. In the Greek mind the greatest thing that you can be a part of was the city (polis - in Greek). Like Aristotle stated: we are all political animals, that is, people who should live in a polis.
Third, foreigners were not treated well. They were second class and Greek rarely ever extended citizenship to foreigners.