I associate the ancient world with times before the end of the Roman Empire (ca. 525 AD). As far as “Modern Times”; I am a bit hazy-perhaps 1492? (Columbus’ discovery of N. America). What do historians consider to be the boundary? And, what defines “modern” times anyway? Is there any concensus? 
I am far from an expert on this but I think it goes from the:
- Ancient World to the
- Dark Ages to the
- Renaissance to the
- Industrial Revolution to
- Various landmarks in the modern world which could be marked anywhere from the early 1800’s to the 1950’s.
All of this is just populist armchair history but I think that is what your question calls for.
Not even trying to locate the academically accepted dates, just giving my own impressions, I equate the end of Ancient Times with the overrunning of the Roman Empire by the Barbarian hordes followed in relatively short order by the rise of the Church and the decline of independent thought.
And even if the Renaissance did start near the time of Columbus’s voyages, Modern Times for me comes with the Industrial Revolution, some century or so later.
No real dividing line. Events associated with the end of the “middle ages” and the beginning of “modern times” are Columbus’ voyage (1492), de Gama’s voyage to India (1497), the fall of Constantinople (1453), the invention of the printing press (1455), the end of the Hundred Years War (1453), the Turkish defeat at Belgrade (1455), the expulsion of the Muslims from Western Europe (1492), and the defeat of the Mongols at Ugra River (1480).
Well, the Ancient Era ends when you learn Construction, Currency, Code of Laws, Philosophy, Horseback Riding, and Polytheism, and the Modern Era start once you’ve learned Flight, Motorized Transportation, and Radio.
Chronos is obviously a Sidmeierist in these matters! 
500 and 1500 AD are not merely convenient round dates but very close approximations of the break between events that fall on one side of the divide and ones that fall on the other.
There were still a few classicist writers operating in the 400-500 AD time period. A few of the Roman Emperors or their generals made serious impacts during that time. But Augustine of Hippo and Boethius are pretty close to straddling the line. Shortly thereafter you have Clovis in France, Theodoric in Italy, the Visigoths in Spain. The Angles, Saxons, and Jutes have moved in and changed Britannia to the English heptarchy, and the Scots are expanding from Dalriada to eventually conquer all of Albany (i.e. what was thereafter Scotland). The beginnings of the Scandinavian kingdoms are separately drawing together, though it will be some time yet before they emerge from legend into history. Meanwhile, in the East Byzantium and Persia have fought to a mutually enervating draw, and Islam is due to begin its expansion over the empires of both. Justinian’s efforts to expand what is now the Byzantine Empire over the West end up largely failures. By 550-600 AD the Medieval world is more or less in place.
Now, 1500: About 1450 comes the end of Byzantine power (1453 is the fall of Constantinople), with three successor states fighting rearguard actions against the Turks for another 100 years. The Seljuk Turks, who had contested Anatolia with Byzantium for 200-300 years, have fallen to the Ottoman Turks, who are en route to an empire stretching from Budapest to Basra to Marrakesh. The five Iberian kingdoms have evicted the last of the Moorish kingdoms, with a matrimonial union of crowns of the three largest in place, to be followed by a union of the kingdoms in due course, and they will shortly absorb the others (though in the case of Portugal it lasts only 60 years or so). England is exhausted from the Wars of the Roses, and France has seized the opportunity to boot England out of its Continental possessions, proceeding towards a unified kingdom from its medieval condition of being divided into small quasi-independent states and possessions of foreign rulers, much like Germany and Italy. The Empire is slowly becoming simultaneously the possession of the Hapsburgs rather than a truly elective office, and meaningless in terms of rule. America has just been discovered, and explorers are setting out to find out what exactly it is. The Renaissance is apace, with its rediscovery of classical thought and art forms, and up in Magdeburg a Dominican theological student is pondering Paul’s Letter to the Romans, with consequences to come that will reshape history.
I always figure that Ancient Times are anything that happened before 1950. Oddly, Modern Times came along in 1936.
Simple, really. “Ancient times” is before I was born; “modern times” is after.
And as far as Medieval Times goes, I was there a year or two ago. Good roast chicken they have there.
Loosely, in the history and art history departments, roughly, “ancient” ends around 400 (with a “late antique” wiggle zone), the “Medieval” period ends 1300 or 1400 or so, in which the “early modern” period starts (i.e. ca 1400-1800), then modernism proper comes around 1800 or so. Dark Ages and Renaissance and Baroque and so forth are being less frequently used.
Yeah, it also depends greatly on what you’re studying. In various history classes, I’ve heard various events thrown around as “The Beginning of the Modern Era”, including the Battle of Trafalgar, the American Revolution (but it seems like sometimes people can’t even agree on when that started or ended), and 1492 (Columbus Sailed the Ocean Blue, as well as the end of the Reconquista).
If I had to arbitrarily pick a date for the beginning of the Modern Era, I’d would peg it somewhere in the mid 1980s, with the birth of a stunningly good looking, charismatic, and intelligent young lad who liked to spend entirely too much time posting to the SDMB and Wikipedia. ahem
Actually, I’d go for 1492, personally. I never quite understood what changed with the Battle of Trafalgar (England went from being the most powerful naval power to… still being the most powerful naval power), and the American Revolution was more of a poltical change of periods than it was any more modern than the few hundred years before it.
As posted earlier, ‘Ancient Times’ trundled along happily in Constantinople until about 1450 and the ‘Dark Ages’ were a lot darker in some places than others - they were pretty gloomy in Britain but things seem to have been a lot brighter elsewhere in Europe.
I would place the start of ‘Modern Times’ around the beginning of the Industrial Revolution, our characteristics are rapidly advancing technology and secular government, although not necessarily democracy.
The usage of toilet paper.
The reason that Trafalgar is so important is that it marks the end of French opportunity to take the UK.
UK v France was a long running dispute for hundreds of years previously, this pretty much finished direct conflict, leaving both to get on with their main concerns.
It effectively cuts the UK off from European involvement, force it to work on world trade.
Its the fall out from the collapse of this colonialist empire, along with the collapse of the mid and eastern European empires that virtually runs internation relations to this day.
This really is a profound statement despite its humorous (I assume) intent. It took me until I was in my late 20’s or early 30’s even to acknowledge the validity of things that came before I was born. Music, movies, world events. Then I began a process of permitting older things to be acceptable as having been direct influences on my life. First I was able to relate to the 1930’s, then the 20’s, then the Turn of the Century, the Victorian Era, the Civil War period, but that’s pretty much where it stops for me. Earlier than that and it’s just hearsay, fiction or “ancient history.”
I would venture a guess that there are less than 10% of the age group Under 30 who can even accept that life began in any meaningful way before 1970. Earlier movies, music, art, “pop culture” and current events are just lumped into one big cloud labeled “Old.” They’re not just irrelevant; they’re invisible.
There do seem to be some youngsters who are not this close-minded to things that preceded them, but their small numbers make this statement of mine so close to valid as to being a major doctrine for Old Folks. It is the basis for the Generation Gap.
Jews were expelled from Spain in 1492. The Muslims were expelled during a period from 1502 to 1614. This is one of those little factoids like the meaning of “Immaculate Conception” that almost everyone is likely to get wrong (until they look it up).
I was born in 1959, and for me the clear cutoff date is February 1964 - Beatlemania. Things were never that same from then on.
I should have been clearer in my previous post. The last Muslim political power was eliminated in Western Europe when Muhammad XII surrendered the Nasrid Kingdom of Granada to Spain on January 2, 1492. There were still Muslims living in Spain after that date, the same way there had still been Christians living in the area during the years of Muslim control.
Yep. As an ancient historian, my studies have ended roughly around ~600 CE (in a Late Antiquity class) and the early modernists start around the Renaissance.
Chariots. The ancient times were over when no one used chariots any more. (Seriously, when the stirrup made a mounted warrior more efficient than a charioteer, around 400-500AD).
Oh, and pants: when men started wearing pants instead of kilts and togas.
As a 17-year-old, I’d have to disagree with you a bit. Stuff before 1940 or so is classified as “old,” sure, but the 50s, 60s, and 70s still hold a lot of interest. The biggest example of this is in music; “classic rock” is incredibly popular. Many iPods in my high school have Frank Sinatra and Dean Martin right next to the Metallica and Eminem. I own close to every Beatles album ever made, lots of Bob Dylan, and even some of the more obscure folks like the Hollies or Donovan, and I’m not at all alone. I just did a quick Facebook search, for instance, and 108 other people at my school have “Donovan” as a favorite artist.
Movies from the era aren’t as overwhelmingly present, but there is a strong feeling from film aficionados that those were the golden days of cinema, and most people are at least conscious of the greatness of those movies. “Bridge on the River Kwai” turns up 78 results as favorite movie on facebook, “The Good, the Bad, and the Ugly” gets 247 results.
I’m going pretty far off-topic here. I think there is general ignorance of pre-WWII culture in my generation. I would guess that the cultural “memory” only extends back one generation - I know a lot about the culture from my parents’ youth, but very little about that of my grandparents. This is normal and isn’t anything to get too alarmed about; the “generation gap” has always existed.
I suppose, then, that the more accurate (but much less funny) answer would be that “Ancient times” is everything before my parents were born; “modern times” is everything after.