Two important elements of democracy in the early Greco-Roman world were:
Democracies in these societies tended to be more or less direct. The most important example is Ancient Athens, were democracy was direct: citizens met in the agora, and discussed public matters, and casted votes.
Democracies in these societies were very limited. Only property holding men, usually those at the top of society in terms of power and wealth, were considered citizens, and had political rights. Democracy in the Ancient Greco-Roman world was not like democracy in our modern world, where most people of certain age are eligible to vote.