The first and most important thing that I have to introduce this answer with is that the majority of Ancient Greeks did not practice democratic government. If you knew this already then I apologise for repeating the information, but it's absolutely key that you do not picture democracy as the default political system of Classical or Hellenistic Greece.

Democracy's history in Ancient Greece has traditionally been dominated by one particular city-state; Athens. It's certainly true that Athens portrayed itself as the progenitor of all other democratic states in Ancient Greece, but the simple truth is that we do not actually know that Athens was the first ancient Greek democracy. If any came before it, we are not aware of them. However, sometimes this has been taken too far the other way- and democracy is portrayed as somehow attached almost solely to Athens and Athenians. In either case, Athens is genuinely important to understanding the history of democratic states among the Ancient Greeks. Precise sources on the constitutional history of Athens are annoyingly infrequent. What those we have suggest is that the process of creating the democratic system was slow, and involved multiple separate reforms which changed the character of Athenian government.

We are told that, for a long time, control over Athens was a position fought over between multiple aristocratic clans and their supporters among the Athenian population. This frequently led to violence. A figure that is alleged to have begun the democratic system in the early 6th century BC, and to have attempted to change this state of affairs, is one Solon. The problem with Solon is that in later periods he became almost a deified figure- he was seen as the ultimate lawgiver of Athens, and almost anything and everything was attributed to him. Commonly he was associated with abolishing debt slavery among Athenian citizens, and for attempting wide-scale reforms. But these were also allegedly rejected. Two other figures associated with the creation of a democratic system are Kleisthenes and Ephialtes.

Traditionally, Cleisthenes is associated with the breaking up of the traditional 'tribal' units of Athens and the focus instead on division by your local areas. The tribes that he created were allegedly redesigned to split up groups across the city's territory, and were also apparently accompanied by the abolition of patronyms (carrying the name of your father) to instead have everyone named by the deme that they came from. (The deme being the smallest unit of this new organisational structure). He also is held to have instituted the election of numerous officials by lot rather than direct appointment.

Ephialtes is a more historically attested figure, and is associated with the erasure of the power of the Areopagus. The Areopagus was, originally, more like the Senate in Rome- a council of elders, exclusively ex-magistrates, who had a large say in the governing of the city. Having already transitioned away from tyrants, the Areopagus had in the early 5th century BC become the centre of Athenian government. However, its power had also begun to wane- Ephialtes is associated with the codification of this, as he took away the Areopagus' rights as a court of law or a body of advisors except in the case of murder and the breach of some sacred laws.

Generally speaking, the movement towards this new system is associated with a period of high instability in Athens and the rest of Greece- many histories of the period talk about a general pattern of inequality in many of the city-states, leading to various kinds of unrest. In the case of Athens, prior to democracy power was relatively funneled towards aristocrats, and they also frequently clashed among themselves leading to civil wars from time to time. Stasis, as ancient Greeks referred to civil war, were considered a particularly traumatic and unpleasant experience. Combined with a widescale poverty problem, which is apparently why so many Athenians had become enslaved due to debt, and it seems to have been quite a powderkeg. Democracy took most explicit power away from aristocrats whilst not eliminating them entirely- Athens never lacked for an aristocratic class, but its influence was only direct over military affairs and a number of religious affairs as well. It also sought, if the reports are accurate, to solve the underlying economic tensions of the period.

However, one thing to add is that Athenian democracy was never, ever still; elements altered over time considerably. For much of the 5th century, Athens is associated with what's often called radical democracy; this is when jurors began to be paid for jury duty, when the Areopagus was defanged, and when democracy was at its most direct. However, this caused resentment among those who considered themselves talented and exceptional, including members of the aristocracy. This is where segments of the population sympathetic to oligarchic systems tended to come from. The latter part of the 5th century BC was dominated by the lengthy Peloponnesian War, and much rhetoric was expounded on portraying the conflict as being between democratic and oligarchic city states (it was more complicated than that in truth). Once Athens began to struggle in the war, the oligarchic elements of the population twice enacted a coup in what's called the Oligarchic Revolutions; the first, in 411 BC, was relatively bloodless and dissolved less than a year later. The second, in 404 BC, was imposed by the Spartans upon Athens' defeat in the war. This was then associated with brutal repression of opponents within the city, and the outbreak of a civil war, which was ended with the defeat of the Oligarchs and the restoration of democracy. However, this democracy was considered to be more moderate than its early 5th century form.

Outside of Athens, we know of a number of other Greek cities that adopted democracy. For example, Syrakuse was a democracy for an extended period, as was Rhodes. However, even the highest estimates that I have seen for democratic cities in the Greek world account for around 1/10th of the total Greek city-states at best. Democracies never represented the majority of states that involved Greeks- the rest were divided between outright kingdoms, oligarchies of various types, federal leagues, and in some cases older notions of community entirely based on wider ethnic groups rather than particular urban communities.

However, ancient democratic government among the Greeks was also very different to many notions of our own. The first being that this was still generally associated with protecting the traditional religious rites of the particular city-state in question; Athens continued to hold the festivals known as Dionysia despite them having originally been founded by one of Athens' previous tyrants. Indeed, the Greater Dionysia is where drama came from. The Greek conception of the spiritual had little room for drama, but what they cared about they cared about a lot. Additionally, many of their legal structures are very unfamiliar to modern democratic systems- not only were most of them decided by lot, and exclusively tied to Athenian male citizens in their exercise, but it also involved the majority of Athenian male citizens at some time or another. By that I mean that the number of jurors and magistrates assigned by lot each year was so high that it would easily mean all Athenian male citizens would have taken part in these roles at least once in their lives. Likewise it was assumed that Athenian male citizens would have performed military service, either as a rower in a galley or by land. Given the relative frequency of wars in ancient Greece, this was not particularly difficult to achieve. But in addition to all this, the Athenian legal system itself was fundamentally different- there were no lawyers, as we would understand them, the juries were much larger than we are used to, there was no arbiter magistrate present at the trials of any kind. Likewise the democratic system did not somehow cause a shift in opinions in other areas- the crimes on the books in many cases seem bizarre to us, as do their punishments, as do the way in which the cases were decided. Speaking ability mattered more than the ability to provide evidence. You could be retroactively punished for a crime, debt could be hereditary, and there was very little in the way of active law enforcement by the Athenian state- in the case of private cases, an award of compensation had to be physically extracted from the party ordered to pay, which could often be as simplistic as turning up at their house and taking stuff.