The Ottoman Empire was one of the longest-standing empires in history, lasting from around 1299 all the way to 1922. Ottoman expansion picked up steam in the reign of Mehmet II, or Mehmet the Conqueror.
This sultan was important because he was the one to conquer Constantinople, the capital of the Byzantine Empire. He did so using a legion of elite janissaries and a massive cannon, called the Basilisc, designed by Orban, a Transylvanian cannon smith.
The next influential sultan was Selim I, who established Ottoman rule in Egypt, claiming the significant city of Mecca to the Ottomans; which meant that the sultans could also claim the title of caliph and were under an obligation to protect pilgrims to Mecca (this will become important later).
Next, Selim's son Suleiman the Magnificent captured Belgrade and the important island of Rhodes. Later, he also conquered the Barbary Coast, which let his pirates attack shipping lanes.
However, the Ottomans' plans didn't all go well. The previous owners of the Island of Rhodes, the Knights of Saint John, were causing trouble, including capturing ships with pilgrims, and even Suleiman's daughter's nurse on a journey to Mecca. Suleiman sent an army to conquer the island, but the plan failed.
Later, Selim II, called The Drunk, conquered the island of Cyprus from the Venetians. However, later, the Ottomans were defeated in the Battle of Lepanto (modren-day Greece) by a Christian fleet. The defeat at Lepanto was in large part to the Ottomans' failure to fully man their ships, and three Venetian galleases, large, heavily gunned sailing ships which splintered the Ottomans' light galleys.
The Ottomans' expansion was slowed in three important places. The first one was the Austrian city of Vienna. The failure to capture this key city meant the end of Ottoman expansion into Europe. The second was Malta, which ended the Ottomans' ambition to conquer Rome. The third was Lepanto, which claimed many of the Ottomans' best sailors and decimated their power over the Mediterranean, a battle that also showed the world the power of heavily armed sailing ships.