Democracy is a form of government in which all eligible citizens are meant to participate equally either directly (through votes) or indirectly (through elected representatives).
So basically in a democracy:
-Elections are free and fair.
-People have civil rights that vary depending on eligibility. (Civil right are Personal liberties that belong to an individual, owing to his or her status as a citizen or resident of a particular country or community.)
-Eligible citizens elect their leaders. (Eligible means suitable or qualified)

The difference between the Athenian democracy and modern democracy is mainly due to the evolvement of the definition of the word eligible.

In the Athenian system, eligible people were: Male citizens. Women, slaves and foreigners could not vote.
In the American democracy, eligible people are: citizens over 18 years old foreigners, and felons serving their sentence cannot vote.
However it is important to note that this has not always been the case, as in the beginning, only land owning white males over 21 could vote. No women, no other ethnicities or foreigners. Which resembles perfectly the Athenian concept of eligibility.

The Athenian democracy was a direct democracy, all male citizens would gather, discuss the issue, and vote on it. However, due to growing societies and the impossibility to gather all citizens in one location, governments came up with representative democracy. Under this system, citizens elect representatives who basically do