The group of Muslims who conquered the area now known as Turkey and expanded into the Southeastern Europe during the Middle Ages were the Ottoman Turks, who were racially Oghuz Turks.
In the early 13th century, an Oghuz Turk by the name of Osman I founded the Ottoman Dynasty in Anatolia.
His descendants expanded the Empire further into Europe and Mehmed the Conqueror was able to capture Constantinople, turning the small dynasty into a major world power.
At its largest extent, the Ottoman Empire stretched from Eastern Europe, all the way to Northern Africa and Middle East.