The Fall of Constantinople marks the end of the Middle Ages and the beginning of a new epoch in Europe. Many Greek scholars moved to Italy, initiating there the development of European Humanism, while the legal succession of Byzantium and leadership of the Orthodox Church transferred to Russian Tsardom and the '3rd Rome' in Moscow. By losing access to the Black Sea Europe was deprived of the land route to India; the search for a new sea route brought about the oversea discoveries of the New World.
When Constantinople fell in 1453 it lead to the destruction of the greatest empire the world has ever seen. The Roman Empire had lasted almost 2000 years it spanned the whole known world, and when a bunch of Muslims destroyed it. The world changed from medieval to the age of exploration. The fall of Constantinople also stimulated the age of European exploration. Basically it helped to create our modern world. How would the world have turned out if Constantinople have not fallen?
The fall of Constantinople also had profound consequences for Europe. Many Greeks and other Balkan peoples, fearing death or forced conversion to Islam, fled westward across the Adriatic Sea to Italy. Many of these refugees took with them vast riches of ancient art and knowledge, helping to ignite the Renaissance.