Osman I founded the Ottoman Empire around 1299 ACE. At the time, the Seljuk Turks controlled most of the Middle East and parts of the Anatolian Peninsula (Modern Turkey). The Seljuks, however, were disorganized and declining. Osman I conquered most of Anatolia and subjugated most of the independent sultanates in Anatolia. His successors eventually controlled all of the territory leading up to the Byzantine Empire, whose capital was Constantinople.
The Byzantine Empire was also weak, recovering from its defeat by the crusaders of the Fourth Crusade. Osman I's son, Orhan, began to seize the territory surrounding the city of Constantanople, both in Anatolia and on the Balkans, modern-day Serbia, Croatia, Slovenia, etc., and modern Greece.
The gradual erosion of the Byzantine Empire was halted temporarily when Timur the Lame (Tamerlane) attacked from Central Asia (Uzbekistan). Tamerlane was incredible brutal. It is claimed that when he invaded Damascus and Aleppo, he ordered each of his soldiers to return with no fewer than two severed heads. After Timur's death, his empire gradually crumbled.
In the meantime, civil war broke out among the Ottomans, ending with Murad II assuming the Sultanate. Mehmet II, Murad's son, brought the Ottoman Empire to its peak when his troops conquered Constantinople.
This marked the final fall of the Roman Empire. Constantinople had been founded on the site of the town of Byzantium in the Fourth Century ACE. The Eastern and Western Empires separated for good following the Death of Theodosius the Great in the beginning of the Fifth Century ACE. The West fell in 476 ACE and the East, which we call the Byzantine Empire (they called themselves "Romans") fell in 1453.
There are several factors that made the Ottomans strong. The first a strong sense of religious mission. The Ottomans were devout Muslims and their Sultan served as both as religious and political leader. They also had a cruel, but effective way of eliminating political opposition. Once a Sultan assumed power, he would have all of his brothers executed so none of them could challenge his rule.
The empire allowed a form of religious freedom. While non-Muslims could not hold government positions, they were allowed to practice their faith. The Patriarch of Constantinople, for example, could still lead the Greek Orthodox Church from Constantinople, which was now called Istanbul and is called that to this day.
There is much more, but this can serve for starters.