I think there is going to be a degree to which this kind of answer is going to be grounded on the history you study - but for the mainstream of western history the modern period is the period that follows the medieval period, beginning sometime around 1400-1500 AD (you'll hear dates as early as 1350s and some dates into the 1500s).

This falls into the four epoch classification of history, in which one can largely divide the history of the west into four broad time periods. Dates are obviously frequently debated and ultimately not that important - very rough ones follow below:

• Ancient History from the beginning of recorded history circa 3000 BC - the developing prominence of ancient Greek City States/dominance of Persian Empire c. 500 BC
• Classical History c. 500 BC - c. 500 AD - from roughly the time of Herodotus to the fall of the Western Roman empire. This time period features the prominence of Hellenism and its descendants.
• Medieval History from circa 500 AD - 1400 AD. This period could end in a few places - some historians choose the Renaissance, which is a pivot towards the art and philosophy of the Classical Era, some choose the age of discovery starting in 1492 with Columbus, there are a lot of other dates that work here. EDIT: /u/Reddit_beats_college (great username by the way!) says that in Europe the Reformation is seen as a pretty consistent breaking point. This is very defensible - one of the hallmarks of the medieval period is the prominence and importance of the Catholic church - first as one of the sole bastions of education/literacy/Roman tradition then as a cultural and political giant. So the if the Medieval period is the age of the Catholic Church - the Reformation represents a major break in history. Personally, I'd say that the rise of natural philosophy in the late 17th century is a good marker of the pivot away from medieval systems of thought, but that's not a widely held opinion.
• The modern era from whatever date you choose to end the Medieval period - present

It's important to note that some historians, especially when writing popular history, will use the word modern differently. I've heard the phrase "WWI is the hinge on which the modern world turns" which refers to the Modern world as being the 20th century world.

This schema outlined above is also very different from the definition of Modernism in art and philosophy, which more specifically refers to cultural movements in the 19th and 20th centuries.

It's also important to note how Euro-Centric this division is. This schema is somewhat dependent on having a worldview that sees one's self as the inheritor of Greek civilization - that's why the rise of Greek City states is important, and the pivot back to supposedly classical philosophy in the renaissance is important. But it also follows patterns of different ways of recording history. The history of the ancient world is relatively impersonal, giving names, dates, actions, but not a lot of "colour." Greeks and then Romans recorded their histories very differently, expounding on personalities, small-scale events, etc. Your mileage may very depending on where you are as well - Italy in 1350 doesn't look nearly as Medieval as England at the same time.

Obviously there was a lot of history happening in other areas of the world that don't fit into the schema at all. Central America not covered here, neither are China, India etc. These would all need different divisions - at least until you hit the modern era. At that point you stop seeing isolated "worlds" of history with only infrequent and minimal contact and start seeing much more interconnection - suddenly the politics of England or the Netherlands can have significant impact in India or Africa.