Classical Period

In the Mediterranean world/fertile crescent the Bronze Age is often said to have ended rather abruptly in a massive calamity called The Late Bronze Age Collapse. Advanced civilizations which had cities, written language, mathmatics and fine products that had stood for thousands of years were swept away or faced major setbacks rather suddenly around the 12th century BCE. Exactly why this happened is a matter of contention and probably not just one thing (crop failure, foreign invaders [often called "The Sea People"], civil unrest and compounding breakdowns were most likely part of it) but regardless the result of which was that society took a fair number of steps back. But things bounced back as a new set of civilizations came about, though considerably different ones from what came before in Greece, the Levant and Italy around 800 BCE or so. Thus began the Classical Period.

The Classical Period is the time of the Greek City States and the Roman Republic and Empire and lasted to about the fifth century CE. What started out as a few minor city states here and there grew into civilizations which would flourish in art, philosophy, engineering, architecture, medicine and more. While many of these states would have Kings at various points in their history, there was also a fair deal of experimentation with various forms of elected government. Even as the classical period would come to an end with the fall of the Western Roman Empire.

The Classical Period is firmly the Iron Age. Iron had been worked before, Meteoric iron had been worked every now and again for thousands of years and occasionally a few skilled craftsmen in the Bronze Age could make a few bits of it even though it was brittle and expensive and mostly used as a sign of status (king Tut was buried with an iron dagger for such reasons). Near the Bronze Age Collapse the Hittites had begun smelting and working iron on a larger scale for more practical purposes such as tools and weapons(and absolutely wrecked a far greater Egyptian army at Kadesh when the iron shields and swords broke the Egyptians' copper weapons like twigs), but it was only after the collapse that Iron Working really became common. Iron has a higher melting point than copper, and the simple wood fired furnaces of the Bronze Age just were not up to the task of processing it in meaningful amounts. If you make a large tube furnace, feed it charcoal, iron ore and oxygen with bellows, you can get it just hot enough to end up with lumps of semi-molten iron goo that can be forged. Regular iron was comparable to bronze in quality for many purposes, but unlike bronze which required two rarer metals that you often had to trade for, iron ore was common as muck. Even so, bronze still found a fair bit of use well into the classical period. After all, the big deal about iron swords is not that they are better than bronze ones, absolutely speaking, but now everyone in your army can easily have one. It would also ensure chariots could now pull better armored soldiers with a lot more ammunition with far fewer risks, but by that moment horses got bred big enough to just carry the fighter.

Notes

This whole deal is a western thing. In China and India and so forth things were going along their own paths removed from all this. Though it should be pointed out that roughly around the same time that Rome existed, the Qin and Han dynasties United China for the first time, being their equivalent of a “foundational” regime that would set political and cultural precedents for centuries to come. Before the west was done with this era though, the Han collapsed and China entered the bloody Three Kingdoms period before being formed into the Jin dynasty.
During this time in Japan the Yayoi period (starting around 1,000 BCE) happened as migrating people from the mainland came to the island chain assimilating the earlier semi-agricultural Jomon peoples and is considered the start of Japanese Civilization as we'd know it.
Writing became more common during this time period. Beforehand in the bronze age literacy was the domain of scribes and a few priests and nobles and existed primarily for purposes of administration first and religious purposes second. In the classical world it was fairly common for men and women who were not at the bottom of society to know their letters and using them much more widely. Probably because the Phoenicians developed the basics of phonetic writing during the Dark Age and it's easier to learn a few dozen symbols that represent short sounds than hundreds of symbols which represent syllables and words. Literacy would decline after the classical period.
Navies emerged as states realized the advantages of travelling over water as opposed to land, and pirates and enemy ships had to accounted for.
Philosophy and political science were at the fore. Every conceivable society was attempted, from Athenian democracy, to monarchies and oligarchies, Rome's landed citizen republic, the theocratic dynasties in Egypt, and the brutal warrior-communes of Sparta. Every society was aware of the growing masses in the cities and the need to keep the populace placated, whether through expanding franchise, brutal tyranny, manipulative privation, brazen demagoguery, panem et circenses, or a combination thereof.
The iconic marble statues were originally painted with colourful paints. The paints have just decayed much quicker than the stone they covered.

The appeal of the Classical Period

Frankly the modern world has a serious Boner for this period. The link we have to the Bronze Age cultures is a bit tenuous at best, but The West sees the Greeks and the Romans as our fore bearers. People like the idea of Greek Philosophers discussing and debating the nature of the world and morality, of Romans forging order from chaos, spreading civilization and building magnificent buildings that stand to this day, Athenian Democracy and Spartan military excellence. Of course, that view is overly romantic and overlooks the nastier side of the period, from slavery to rampant xenophobia and sexism (especially with the Greeks who only had a few men voting to the point that the "democracy" was little more than an oligarchy of a few) to the fact that this could be a rather brutal period with a lot of pig headed stupidity at the time. Many people have tried to emulate the better notions like bureaucracy and checks on power and build on them.

It helps that we actually have a fair bit of information about this time from first-hand accounts. Historians have to parse through a smattering of tablets and decorations on walls for the Bronze Age, much of which they can't read. In this period we have a good index of this time period, from Greek poems and plays to biographies and histories. Even if said writings aren't very objective, they make filling in the blanks a hell of a lot easier and gives us insights into a lot of different people, which means we have a lot of characters to get insights on how people got along back then.

Finally, there is something of a mix of the modern and the ancient in the Classical Age that you don't get in the medieval period. In Rome people lived in apartment blocks, had sewers to take away their filth, had theaters and coliseums to keep them distracted, and (if they had said status) had a conception of their role in society as citizens with legal rights and listened to political rhetoric and heard satire that's not too different from what someone in a first world country would hear as opposed to how a medieval peasant or knight would. Bob-every-Roman puts in a hard day's work selling olive oil so that when game day comes around he can go down to the arena with his bros, drink wine and bet on the gladiators.

Mind you, this stuff existed in a world where slavery was normal and unremarked on. Romans unironically complained about all the slaves taking their jobs, and some portion of them actually had a point: the modest farmers who got screwed by big landowners and aristocrats who bought them out while they were at war and then used their slaves to profit off of the land ended up creating a big urban poor population that was the cause of many of the late republic problems. Having criminals fight to the death was seen as prime penology (and good entertainment) and people sacrificed sheep so that next years' grape crops would yield a prime vintage was a regular part of religious lives.