The Athenian democracy was insane. The city's leaders (ten archons a year, 500 council members who served in 10 shifts, etc.) were drawn by lot. The only elected positions were generals and people who oversaw the treasury. These people could only serve once for one year (again, except for the generals). This served to keep the government week and keep the power in the hands of the assembly which decided on every single issue of state. How to raise money, how to spend it, what strategy to take in war (the generals made the decisions once the armies were sent out but the assembly decided where to send them), who would lead the big religious festivals that year, etc. This assembly was made of every adult male citizen in Athens.

Does this sound like any democracy you know?

What we consider democracy, Athenians would have called oligarchy. We elect members to sit in assemblies (parliaments, congresses) and make decisions and sometimes powerful executives for the country. In ancient Greece an oligarchy was a place where a council oversaw affairs instead of an assembly (some of those councils were elected, others were not).

If you want more, you can check out Aristotle's "Constitution of the Athenians" and "Politics" especially books 4-6.

Many governments purposefully avoided the Athenian model (the US, for example) because it was risky and willful. The democracy was pretty bad for the Peloponnesian War for example. Most modern governments modeled themselves on Rome, more than anything, and any connection between the Roman constitution and Athens's is sketchy.

Ancient democracies were modelled upon Athens, mostly because they were in alliance with Athens.

For "modern" democracy it's hard to say. First which state had the first democratic government? San Marino claims to be the first democracy, but I personally can't really say if it is or not, because I haven't really delved into history of its government.

Based on what I researched (I have a masters in Political theory and my masters thesis dealt with relation between democracy and republic at US constitutional convention). I'd say that the first really truly democratic government was the government of colonial Rhode Island, more specifically the document that unified cities of Portsmouth and Newport. The document clearly state, that:

the Government which this Bodie Politick doth attend vnto in this Island, and the Jurisdiction thereof, in favour of our Prince is a democracie, or Popular Government; that is to say, It is in the Powre of the Body of Freemen orderly assembled, or the major part of them, to make or constitue Just Lawes, by which they will be regulated, and to depute from among themselves such Ministers as shall see them faithfully executed between Man and Man. source

I unfortunately haven't found any primary source, that would show a clear link or influence of Athenian democracy on this agreement. But the fact that they named it democracy, instead of republic or commonwealth, both of which were more highly regarded than democracy, does seem to showcase, that a) they knew about Athenian democracy, b) didn't think that it's a bad form of government.

This kind of outlook on democracy, is (as far as I can tell) in minority, as most political philosophers saw democracy as deeply flawed and more like a rule of mob, compared to the highly regarded status it has today.

Colony of Rhode Island remained "democratic" even under newer constitutions, but those didn't specifically name the colony a democracy.

Next positive outlook of democracy can be found (or at least I found) in John Wise's A Vindication of the government of New England churches, where he defends the idea, that democracy is the closest government of the early Christian communities. But he doesn't really found any of his definitions or reasons for democracy on Athenian historical example.

I would again point out, that most of the time democracy wasn't highly regarded. For instance John Cotton specifically assures Lord Say and Seal, that Massachusetts Bay wasn't democracy. And probably the best example on how democracy was seen in those days can be read in Federalist #10. Another example is John Adams' A Defence of the Constitutions of Government of the United States of America. Now that I'm re-reading it, I see that he does goes into description of "democratic" governments (of his time), which also includes San Marino. He also references Voltaire, who was known proponent of democracy. Yet he still comes to the similar conclusion as Madison does in Federalist #10, that democracy can only work on smaller territory, and that republic is a better form of government, when government is divided on executive, legislative and judiciary.

On the end I would also like to point out that there is a huge difference between democracy and republic, which are in modernity seen as pretty much the same thing. Democracy can only be, either government where all the people (that have power to participate in politics) meet and decide, or where the people name their delegates that represent them, but can't accept anything without direct consent of the people (see: Rhode Island's constitution of 1647). In republic on the other hand people elect their representatives who are then free to pass any law they want, based on the indirect consent of the people, which the representatives got with election.

One of the things to remember in this case, is that Athens was not viewed as the most important/interesting city state of ancient Greece until Grote's work - so until 1850 or so, Sparta was the go-to example and held to be ideal.
Some authors, though, like the German Christian Meier, would tell you that the "democratic legacy" of ancient Greece (Athens specifically) can be traced straight on through all of European history - however, the texts read in the "middle ages" pertaining to the Athenian democracy weren't read for that purpose, they were usually kept as examples of good oratory, and were further muddled by added scholia when they arrived in Europe. What really kicked off the interest in Athenian democracy was the rise of modern representative democracies and George Grote's work. The only example of a truly unconnected democracy can be found in the cantons of Switzerland where they have proceedings very similar to the Athenian Ecclesia (Athenian Democracy and Ours), but mostly the modern democracies are built more on Rome than Athens, as qsertorius explains above.