The early Triassic Period, which followed the PT Event, the greatest mass extinction in Earth's history, and the Paleocene Epoch, which followed the KT Event, the most well-known mass extinction, wherein, among many others, the dinosaurs were wiped out. note Well, the large ones, anyway. The smaller theropod dinosaurs live on to this day in great numbers. We call them 'birds'.
Humanity may have come very, very close to extinction, multiple times. There's the Toba catastrophe theory. Shortly put, our numbers were reduced to around 10-15 thousand individuals due to a volcanic eruption around 70,000 years ago. In addition, humanity can be genetically traced back to a mere 50 women. It's possible that there were many more others alive and we simply didn't inherit their genes. Regardless, there are some evidences (such as the Toba theory above) that the human race did go through a population bottleneck of some kind very long ago, either because of a major global disaster or simply because a small group managed to pass their genes along to the present population while the others failed to do so for more mundane reasons.
Antiquity
The time after the Bronze Age Collapse, which covers the Greek dark age of 1200-750BC could be argued to fall into this category, with a widespread collapse in Mediterranean Bronze Age civilization, including the Mycenaeans note proto-Greeks, Hittites, Canaanites, and Minoans, and also a greatly weakened Egypt. It's said to be worse than the collapse of the Western Roman Empire and often cited as the worst case of societal collapse in human history. It inspired The Iliad and The Odyssey, and some of the early books of the Bible. The weirdest thing is that scholars have almost no clues as to what happened. Warfare is often cited as the main culprit, with records indicating mass raids by the mysterious "Sea Peoples", but more evidence points to that and fragmentations within those societies, famines, epidemics and natural disasters. Nobody really knows what happened, however (one of main going theories is that there is no main culprit — the civilizations could have endured any one disaster, but not multiple at the same time, especially not when they were feeding into one another).
Complicating matters is the very nature of the Bronze Age. Bronze working requires both copper and tin. Copper is fairly abundant, but tin is rarer, and seldom found in the vicinity of copper, so it needs to be imported. For a long time these long distance trade routes created a sort of golden age of commerce in the Eastern Mediterranean, but if something happened to the trade routes, then various societies would be cut off from their main source of metal, which means no more tools, farm implements, weapons or armour, which makes it harder to grow crops, fight off invaders, build ships, and all of this is greatly destabilizing, which further disrupts trade... One of the reasons why the subsequent Iron Age was the age of iron is that iron is much easier to get, simplifying supply lines.
Dark Ages
The period of European history between the fall of Rome and the rise of Charlemagne is sometimes called "The Dark Ages", and said to be characterised by doom and gloom. This is somewhat of a misnomer created by Renaissance scholars, who were such Ancient Greece and Rome fanboys (well, mostly fanboys) they didn't think a period without classical knowledge could be anything other than post-apocalyptic (Petrarch, the father of Renaissance humanism, said "What, then, is history but the praise of Rome?"). In truth, a lot of classical knowledge was lost, but it was more on the order of "we can't build classical-style ampitheatres anymore" than "devastated world of crap". It wasn't that horrible most of the time, save for Viking and Magyar attacks. And the constant border wars that came with the political fragmentation — not, of course, that the last centuries of Rome hadn't been marked by this sort of thing — and the rise of the armed aristocracy. And long wars, albeit ones which didn't really affect the wider population too much. And serfdom, which was always a bit of a downer if you weren't a noble or a clergyman (but is a big step up from the slavery system of Rome). Nowadays, the notion that the Dark Ages were really dark is considered dated history and has been debunked by historians.
Classical civilization continued unabated in the Eastern Roman Empire (the "Byzantine" Empire). "The Dark Ages," a misnomer as it is for western Europe, did not exist as far as the eastern Romans were concerned. In fact, Greek scholars steeped in classical learning migrated to western Europe after the fall of Constantinople and made a significant contribution to the Renaissance.
The Dark Age of Europe should be understood more along the line of "we don't have many documents from that time period" (which doesn't necessarily means people didn't write anything down, just that if they did most of it was lost to history) than "things were horrible". It's a "Dark Age" in the sense that we're "in the dark" about it.
Pre-Columbian Central America can count as one after the fall of the Mayan civilization. The Yucatán peninsula was once a vibrant agglomeration of culture, learning, and great metropolitan city-states, well versed in sciences from engineering to astronomy. From around 250 A.D. to 900 A.D., they erected great temples and statues and formulated a complex writing system, forming a civilization that would put most of Europe to shame. Then, for reasons not yet completely explained, they entered a period of massive decline, the main political entities failed, dynasties collapsed, and many of the once-great cities were abandoned. The most probable cause was environmental degradation and a huge food shortage, which caused most of the Mayans to flee to greener pastures. By the time the Spanish conquistadors arrived, the region was a shadow of its former self.
Despite this, the Maya didn't really vanish in what English-speaking archaeologists currently call the Postclassic period, 900-1500 CE- some of the Mayan city-states of the Classic period collapsed; but others continued in what is currently Oaxaca, Cholula, & Yucatán (e.g. Uxmal in Yucatán was a Mayan center through to the mid-1500s). The Maya retained and expanded their literature during this time; further developed metallurgy; and interacted with the Aztecs & many other groups. Then the Spanish showed up; decimated the Maya by disease and war; and burned every Mayan book they could find. Some Maya survived all of that and preserved their languages even as their writing system & most of their literature was destroyed. There are more than six million Maya alive today.
Middle Ages
Europe and Asia after the plague of 1347-1350, otherwise known as The Black Death, the Great Mortality, and the Year of Annihilation, is as close as human history has ever gotten to After the End. The continents' populations were at a high, with population growth slowing down in the preceding decades as the land was stretched to its sustainable limits. When the plague hit, it struck at a time of relatively high population density and malnutrition, the death rates ranging from 15% to over 60% from area to area. Though very little of the population - 10% is a common figure - lived in towns and cities, these were the places that got the worst of it and many were almost totally depopulated as people died or fled. In Europe, it dealt a heavy hand to the nobility, the Church, and higher learning in general. The relative scarcity of the peasantry meant they were in a much better position to negotiate their working conditions and wages, at the expense of the nobility. Pre-plague, most literate people were clergymen and nearly all clergymen were literate; clerical literacy did not reach pre-plague levels until the sixteenth century, with the advent of printing and a concentrated programme of Clerical and general education. Furthermore, the devastation of the plague and the resultant social upheaval cultivated rebellion, religious fanaticism, and anarchy (such as the anti-Semitic, anti-authority flagellants). Even before the plague, the 14th century had been a crappy time to be alive and European what with the Hundred Years' War (which wasn't too bad, actually), the Mongol Invasions (which actually were rather bad), various natural disasters, and the years of crop-devastating heavy rains (rain encourages fungi, which at its best reduces your crop yield and at its worst kills you when you eat your bread) that preceded the plague in some areas.
One of the (many) factors that contributed to the great success of the Muslim Conquest is that it happened After the End of the war that exhausted both Byzantium and Sassanid Persia. By that time, Persia was a mere shade of its former glory, they couldn't even mount effective resistance against the newly-emerging bands of Arab raiders, and while the Byzantines survived, they were reduced to a Vestigial Empire, albeit one that hung on and even experienced resurgences for the next eight centuries.
Age of Discovery / Early Modern Period
The Americas after the Columbian exchange, which brought a slew of European diseases to the New World, including smallpox, influenza, and plague.
It was no coincidence that the Pilgrims chose Plymouth as the site of their colony. It turns out Plymouth was an empty Indian settlement. The Pilgrims arrived just shortly after a massive pandemic of smallpox (or possibly plague—the evidence isn’t 100% clear) had swept the area, which wiped out over 90% of the Native American population. Instead of trying to eke out an existence in the untamed wilderness, they took over the empty Indian settlement and its surrounding farmland and infrastructure, even going as far as opening the recently-dug graves to search for anything useful to salvage. In addition, what little remained of the local population was in no shape to seek any fights with the newcomers. Some people have speculated that the American Union's fascination with post-apocalyptic scenarios has its origins in these first, formative years.
Plymouth wasn't the only such location. We now understand that the American continents were teeming with people, that there were multiple, vibrant civilizations. Plague after plague swept through, literally and repeatedly decimating populations. That we know so little about many native cultures and histories is due in no small part to the fact that 2/3rds of their populations died. Everything since the arrival of Europeans has been After the End for Native Americans.
There is some controversial evidence that some of the well-known Great Plains tribes (the ones that have captured modern fascination) may have been formed after Columbian contact. If true, it almost certainly means they were formed out of the remnants of civilizations laid low by plague. It would also mean that when they acquired horses (imported from Europe), they became the equivalent of the biker gangs from Mad Max, or at least the people they descended from would view it that way. In any case, survivors of sedentary civilizations almost certainly joined Plains tribes even if they didn't form their own. Think about that. Going from being a farmer living in the house your ancestors built to being a nomadic hunter-gatherer warrior.
A widespread viewpoint in 17th century Russia. The century started with several lean years in a row, caused by abnormal colds. The resulting famine killed two-thirds of the population. The tsar was overthrown and a decade of coups, civil wars, and foreign invasions followed ("Uncertain Times" or "Times of Trouble"). The order was somewhat restored under the newly elected Romanov dynasty, but life wasn't easy. When schismatic priests in the 1660s declared that the end of the world is happening and the country is ruled by the Antichrist, people tended to agree.
And as the people of Russia have somewhat got used to the church reform... came Peter the Great. Who had promptly introduced his subject to such horrible things as foreigners, shaving, tobacco, foreigners, regular army, "Hungarian" (aka European) clothing, "assemblies" (parties for aristocrats with royally mandateddebauchery, drunkenness and all sorts of shenanigans)... and did we mention foreigners? Add to this quite a few wars and you can understand why so many people believed that the end of the world has already happened. Peter was called the Antichrist repeatedly. His abolition of Russian Orthodox Patriarchy (and requisition of church bells to be melted and made into cannons) may have helped the sentiment a wee bit...
In modern times, there are a lot of Russian cities completely abandoned due to their industries no longer being sustainable. However, more eerie and fitting those tropes are the "half-abandoned" cities in the same category. The people living in these run-down monuments to the fallen Soviet Union often talk about the world as if the end of the Cold War was in fact the end of civilization. Much of the local "industry" revolves around salvaging cities and buildings that are in fact abandoned, and trading what little they do still produce in exchange for supplies. Public spaces have been converted into subsistence farms and any machinery still functional has been cobbled together from what they can find. But the Lenin statues, Red Stars, and all the hammers and sickles are all usually maintained almost religiously and they don't take kindly when authorities from the new Russia come and pay them a visit.
Thirty Years' War was seen by contemporary sources as a truly apocalyptic war: in some parts of Central Europe, the death toll was proportionately worse than in the World Wars, central government and authority collapsed in some areas, and armies reduced to nothing more than pillaging mobs that looted anything not nailed to the ground and burning anything that was. This was compounded by climate change, which reduced harvest yields, and inflation which made it hard for anyone to afford what little there was. The social trauma of the Thirty Years War is what led to the end of large scale religious conflict in Europe.
19th Century
The arrival of rinderpest to Africa in the late 19th century was the moment when the continent truly became Darkest Africa. An infectious disease that affects cattle, rinderpest arrived in 1887 when Italian soldiers accidentally brought it to Eritrea, and it spread like wildfire through the cattle of sub-Saharan Africa, who had no immunity to the disease. Pastoralists lost their entire herds, farmers lost their work animals, and the resulting famine killed one-third of Ethiopia's population and two-thirds of the Maasai in East Africa.note The disease was confirmed eradicated in 2011.Worse, without the grazing animals, the grasslands turned to bush — ideal habitat for the tsetse fly, a notorious killer that exploded in population and left millions more people and livestock dead of sleeping sickness. Conservationists who arrived after the fact dubbed the tsetse fly "the best game warden in Africa", rendering large areas inhospitable for long-term human habitation without serious human intervention to remove the fly's habitat. The devastation left by the tsetse fly made Africa easy pickings for the European powers, and to this day, it is a major public health hazard on the continent that plays a large role in its lack of development and poor living standards.
20th Century
China after the Xinhai Revolution, with No More Emperors and all. Simultaneously a period of great political and social chaos and coupled with huge progress and massive muck-ups in culture and economy.
Today, Hiroshima and Nagasaki are thriving... but this is because the bombs used were for shock and awe rather than sheer destructive power, were detonated in the air (and so didn't leave significant fallout), and were very small in nuclear terms (15kt for Little Boy and 20 for Fat Man).
The fire-bombing campaign was more on this order. In one particularly spectacular attack, 40% of Tokyo was razed in a single night. It killed at over a hundred paces and left virtually nothing in its wake.
Albert Einstein, who saw the dawn of the Atomic Age and the beginning of the Cold War in the 1940s, once made the following grim prophecy: "I know not with what weapons World War III will be fought, but World War IV will be fought with sticks and stones." His point being, of course, that if the arms race did or does lead to a global nuclear war, humanity would be driven back to the stone age by the destruction.
Other / Multiple Periods
The entirety of human history has happened after the end... of the dinosaurs. That's right kids, we live in a post-apocalyptic world. The dinosaurs themselves lived after the end of the Permian-Triassic mass extinction, so you could say the dinosaurs themselves lived in a post-apocalyptic world.
There is evidence that the entire solar system is cobbled together from chunks of other solar systems blown apart by supernovae which, given the age of the universe, were probably made from other systems and so forth. In fact, the Milky Way itself is probably a new system made up of older galaxies which in turn were made up of the remnants of quasars and so forth and so on right on back to the Big Bang.
In reality, the worst "cosmic" tragedy to have any feasible chance to affect mankind at all is localized space junk affecting just our tiny planet, and even then, it will in all likelihood be detritus from objects we ourselves put in orbit coming back to bite us in the collective ass. Hazardous microscopic trash hurtling through stable orbit above us at ludicrous speeds, destroying at random some satellite we put up, and chunks thereof exponentially multiplying the threat to further man-made craft sent into or passing through orbit, eventually "grounding" us and our further ambitions and creations entirely, as per the anime Planetes. Possibly... The plots of Deep Impact, Armageddon (1998), etc.? Unlikely. Galactic or universal catastrophe ever reaching out as far as to affect anything still recognizable as a continuation of humanity? Impossible. Even if we were in fact obliviously living After the End of a new, latest Big Bang, even if it had already extinguished 99% of the celestial bodies whose delayed, ancient light is reaching us just now, neither humankind nor any other current species on Earth would ever be long-lived enough to still endure to a time when such hypothetical destruction arrives to any point close enough to affect us. Fact.
Our Universe is in a sense living After the End. Eternal Recurrence-like ideas as those commented below apart, the very early Universe just after the Big Bang was one very different, exceedingly dense and hot where the four fundamental forces of Nature were merged into just one. Whatever had existed by then would have faced said "superforce" dividing into the ones we know and the Universe cooling and becoming less dense as it expanded, with cosmic inflation if it had happened being the "coup de grâce" for that. There's in fact strong evidence that suggests the onset of cosmic inflation was the Big Bang, that would have erased all the properties of such previous Universe.
One cosmology theory holds that the universe continually expands (Big Bang) and then eventually contracts (Big Crunch), destroying everything before another Big Bang starts the cycle over again. If the theory is correct, literally everything in the (current) universe is "After the End" of an untold number of previous universes.
Seems to be ruled out in favor of the Universe expanding into oblivion (Big Freeze). However after a very, really long time, a new Big Bang could appear from that nothingness (Roger Penrose has a similar idea on his book Cycles of Time: An Extraordinary New View of the Universe). That short of some theories that suggest the formation of a black hole could give birth to a new, independent Universe via a Big Bang.
. Shortly put, our numbers were reduced to around 10-15 thousand individuals due to a volcanic eruption around 70,000 years ago. In addition, humanity can be genetically traced back to a mere 50 women. It's possible that there were many more others alive and we simply didn't inherit their genes. Regardless, there are some evidences (such as the Toba theory above) that the human race did go through a population bottleneck of some kind very long ago, either because of a major global disaster or simply because a small group managed to pass their genes along to the present population while the others failed to do so for more mundane reasons.