This is very large topic, I could go on for hours, you should read some basic ideas from Plato ("Politeia"), Aristotle, the "golden era" of Pericles in Athens...I think Wikipedia says a lot though, and it should be enough for homework:
http://en.wikipedia.org/wiki/Athenian_democracy
http://www.historyguide.org/ancient/pericles.html
Demokratia (in Greek) = demos (citizens) + kratos (state, power)=power lies in citizens.
A basic difference with nowadays though is that in Ancient Greece there was a direct type of democracy, meaning citizens,(only men and only free citizens,not slaves) could directly participate in the discussions of the assembly (lets say a type of parliament). They ruled themselves.This is not possible today because of the increased population, so we elect representatives.
Nowadays we don't have slaves and women do vote, but don't we have our own categories of modern excluded citizens?You should think about that and compare.
Whatever you read keep in mind that democracy in ancient Greece may not have been perfect, but it was adjusted to contemporary way of thinking. So things that we may think are wrong now, may have been completely accepted at that period of time.