Most historians understand that periodization is often arbitrary, artificial, and imprecise, even though people still refer to distinct periods of history as a matter of convenience. In reality, no one has ever woken up and said, for example, "Goodbye Bronze Age! The Iron Age starts today!" As you probably know, history also involves continuity, which doesn't always fit well with the distinct divisions that you've described. :)

As /u/ScipioAsina pointed out, periodisation is arbitrary and artificial. Often there isn't one event that can be used as a "turning point", and definitions often shift as time progresses. While the fall of Rome may seem a good indicator for Ancient History to end, there are other important events that historians use, e.g. the baptism of Clovis (~500), the coronation of Charlemagne by the pope (800), or the large migration period between 500-700AD, as well as the spread of Islam in the Middle East and Africa.

Does modern history start around 1500CE with the age of discovery?

The discovery of America by Columbus (1492) is one of several events that mark the end of the Middle Ages and the beginning of Modern History. Others include the invention of the printing press (1440), the fall of Konstantinopel (1453), and Luther's 95 Theses (1517). Some historians even take the beginning of the French Revolution (1789) as the beginning of Modern History, whereas others see the revolution as the turning point between Early Modern History and Late Modern History.

Source: Müller, Harald (2010): Mittelalter. München: Akad.-Verl. (Akademie Studienbücher Geschichte).

[broken into two answers because of length]

I agree with the two comments above noting that the divisions of historical time periods are largely arbitrary; however, we can still identify certain characteristics that distinguish one period from another. Identifying those characteristics, though, is a difficult task, because it depends on what specific elements of society you are looking at. For example, if you are considering the history of technology, there are a couple major turning points you could consider to divide periods, such as the appearance of the stirrup in the medieval West in the late 10th century or the invention of the printing press in 1452. But if you were considering something from a cultural point of view, such as perceptions of the holy, you would need to take into account different events/themes to distinguish between your time periods.

What this means is that the periodization of history is largely artificial and dependent on what discipline of history you're undertaking. Are you a political historian looking at the development of the institutional state? If so, then you would probably consider the Early Modern Period (roughly 1500-1750) and the institutions and administrations that developed over this time in contrast to earlier or later eras. What if you're a historian of the economy? Well, then your individual understanding of the divisions of history and how and where things either changed or remained the same will vary.

In general, though, historians have agreed on several key terms to identify periods of history. What follows below is purely based on Western history and does not apply necessarily to other areas of history. I am also avoiding including modern history, because that is not my specialty and a modern historian should speak about the periodization of the 18th-21st centuries.

The two most basic questions an historian must consider when trying to divide history into periods are: 1. When are there moments of change? and 2. What are the moments of transition? Change rarely happens instantly, and instead of seeing drastic differences between one period and the next, we instead find lines of continuity connecting Western civilization from one period to the next (and to the next and the next sometimes).

Historians also need to consider place when they are dividing time into periods. Often, the geography and actual landscape shapes culture to a large extent, although following this line of thought can lead down the dangerous road of geographical determinism. But! Keeping place in mind can provide context for points of continuity and change. For example, if we look at the Mediterranean world over a long period of time, several centuries, (an approach historians call the longue durée), we see shared cultural elements that transcend the political boundaries. For more information specifically on what this looks like in the Mediterranean, check out Fernand Braudel's The Mediterranean at the Time of Phillip II. It is an old book, dating from the 1920s, but it will provide you with a good sense of how scholars have been wrestling with the problem of periodization for a century now.

Here are the rough dates and periods defining Western history from the 2nd millenium BC (in Greece) through the early modern period. Keep in mind that these are rough estimates rather than strict guidelines. I am avoiding the general terms of "ancient" and "medieval," because periodization is complex issue and it is better to think of the following periods as times of slow transition rather than moments of abrupt change. We're dealing with hundreds of years here.

Ancient World: I'm not going to give dates here, because as you noted in your question, the ancient world covers not only a vast amount of time, but it also covers multiple regions of the world: Mesopotamia, Egypt, Greece, Anatolia, Rome, Carthage, Gaul, etc. It is hard to really nail down exactly what was included in the ancient world and when it began or ended. The idea of the ancient world also included smaller categories, such as the Greek Iron Age (rise of Minoan and Mycenaean civilizations, 1500-1000 BC), the Greek Dark Ages ("collapse" of Mediterranean civilization, 1050-800), Archaic Greece (resurgence of written texts and a period of centralization, 800-470), Classical Greece or the age of Pericles, Socrates and Plato (5th and 4th centuries BC), the Hellenistic World (characterized by the spread of Greek culture and language from Egypt to India after the death of Alexander the Great, 323-??? BC see here for info on the confused dating of the end of the Hellenistic world), the rise of the Roman Republic (509-27 BC) followed by the rise of the Empire (overlapping with the Republic, defined by the expansion of Rome rather than the presence of an Emperor). But then, as you note, the Roman Empire "falls" in 476. Yet, it's not that simple. Remember, we are dealing with times of transition rather than moments of distinct, swift change.

When we are speaking about the ancient world, we are largely referring to the time before the Middle Ages; but, when exactly do the Middle Ages start? There are two terms you need to know first to understand how scholars understand the transition from Roman world to Medieval. First, Late Roman. Second, Late Antiquity.

Late Roman (150-550 AD): I would characterize the Late Roman world in the West as beginning the second century and continuing until about 550. In my opinion, the distinction between Late Roman and Late Antiquity comes in the way scholars apply these terms. Late Roman often refers to the political changes that characterized the Empire. I have chosen to place the end around 550, because after the 6th century, any hopes of re-uniting the Roman Empire, East and West, are largely set aside as the Eastern Roman/Byzantine Empire have other concerns to deal with. The Late Roman world includes by the rise of Christianity, the Crises of the Third Century, Diocletian's division of the Empire, Constantine's moving the capital to Constantinople, the migration of the Germanic peoples into the Empire, and the sacking of the city of Rome by the Goths. However, I would argue that Late Roman should be extended beyond the fall of the empire in the West, largely because I don't think people actually perceived the Roman Empire to have fallen until after the Emperor Justinian's efforts in the 6th century to reconquer Italy (including Rome), North Africa, and parts of Iberia. For more information on Justinian I and his place in defining the end of the Roman Empire in the West, see volume 2 of Bury's History of the Later Roman Empire from Arcadius to Irene (2013).

Late Antiquity (300-650/700): I would argue that Late Antiquity differs from Late Roman, because Late Antiquity is inherently seen as a time of transition out of Roman culture and into a Christian culture. I mention the Christian culture, because I believe the rise of Christianity is one of the defining shifts from the Classical world into the Medieval. The term Late Antiquity was first introduced to English-speaking scholars with the publication of Peter Brown's 1971 World of Late Antiquity. Even forty years after its publication, Brown's understanding of the period from 200 to 650 remains a foundational element in conceiving the "end" of Roman culture and the "beginning" of medieval. (I have put "end" in quotations, because I'm not entirely certain elements of Roman culture ever actually ended. If interested, ask in a comment.)

Late Antiquity is a useful term, because it forces us to reconsider the "fall" of Rome. One of the best summaries of recent scholarship in the field of Late Antique studies is Chris Wickham's The Inheritance of Rome (2010). In the opening chapters, Wickham explores how Christianity shaped the Late Roman political system, and he breaks down when are key points in the "collapse" of the Roman Empire. Perhaps most interestingly, Wickham includes many details about the Late Antique Mediterranean economy to show how the Germanic migrations, particularly the Vandal conquest of North Africa (which had the most grain exports), broke down the Empire on an administrative level rather than through the sacking of cities. At the same time, though, he makes it clear that people of the Roman Empire did not anticipate a collapse. Wickham also notes that the administrative and governmental institutions of the Roman Empire did not vanish, sometimes until the 7th or 8th centuries.