The expansion of the Ottoman state from a tiny principality in Anatolia into one of the largest empires of its time had a profound effect on early modern Europe, much of which endures to this day.
Sultan Mehmet II conquered Constantinople in 1453, marking the final end of the eastern Roman (or "Byzantine") Empire. Angered by the memory of the Crusades, and by the ongoing Reconquista of Muslim Andalusia by Spanish and Portuguese Christians, the Ottomans declared an embargo on European trade with Asia. European traders sought alternate routes to the Far East, beginning the Age of Exploration which led to the discovery of the Americas and of the Pacific Ocean.
By that time, however, Ottoman rule in southeast Europe was already well established. The very word "Balkans" to describe the region between the Adriatic and Black seas originated with the Turks, who named it after Balkh (Bactria) -- a Persian province in what is now northern Afghanistan. Their expansion led to repeated wars with the Austrian Habsburgs; in 1529 and again in 1683, they laid siege to Vienna itself. The second time, they were turned back only by a massive alliance (instigated by Pope Innocent XI) of the Papal States, the Holy Roman Empire under the Habsburgs, and the Polish-Lithuanian Commonwealth under Jan III Sobieski.
The Ottoman navy ended Venice's maritime dominance of the eastern Mediterranean, and rolled back Spanish expansion into North Africa after the Reconquista. In so doing, they preserved a safe zone for Andalusian Muslims and Jews (known as Sephardim) who had been expelled from Spain and Portugal.
The Ottomans also conquered Christian kingdoms in western Asia, including Armenia and Georgia, and sent Sufi missionaries into the Caucasus. This was the beginning of a centuries-long rivalry with Russia for dominance in the region. In the late 17th century, Russia joined the post-Vienna European alliance against the Ottomans, in what became known as the Great Turkish War.
The dramatic successes of the Ottoman military inspired European states to copy their organization, equipment, and tactics. European rulers recognized the high efficiency and morale of the Ottoman Janissary corps, and began recruiting and training their own professional armies instead of raising ad-hoc levies of conscripts or hiring mercenaries. The Ottoman infantry were one of the first to be issued with personal firearms as standard equipment, instead of relegating them to skirmishers or other auxiliaries. The Ottomans even organized the first modern military bands.
The Ottomans united dozens of nations, languages, and religions under a common political and economic system. Peoples, goods, and ideas flowed freely across the Empire. Greeks, Armenians, Jews, Turks, Arabs, Copts, Assyrians, Berbers, Kurds, Albanians, Croats, Serbs, Bulgarians, Romanians, and many others lived and worked alongside one another, with a degree of peace and prosperity no European power had attained since the height of the Roman Empire.
But unlike the Romans, the authors of this Pax Ottomanica were always seen by Europeans as intruders into what they had grown used to thinking of as "Christendom". As recently as the mid-20th century, central European mothers invoked "the Turk" as a bogeyman against children who showed a propensity to wander too far away from home. As refugees from the present chaos in former Ottoman provinces like Syria, Libya, and Iraq continue to pour into Europe today, the anxiety once engendered by "the Turk" is filling European hearts once more.