As most reports on the group have noted, Khorasan refers to a historical region that encompassed northeastern Iran, southern Turkmenistan and northern Afghanistan. It was established as a region by the Sasanian dynasty, the last Iranian empire before the rise of Islam, at some point in the third century. Its name literally means "The Land of the Sun," a reference to its eastern location.
After the region was taken over in an Arab conquest in the seventh century, Khorasan became a part of the Umayyad Caliphate, and with that, part of early Islamic culture. Notably, a widely discussed (though disputed) Hadith speaks of how "black banners will come out of Khorasan" in the end times.
Will McCants of the Brookings Institution notes that the prophecies derive from the eighth century Abbasid revolution, which began in Khorasan and saw the end of the privileging of Arabs over non-Arabs in the Islamic empire.
Over the years, the Khorasan region had a fractious history, and was eventually swallowed up by a variety of different states. A part of Khorasan eventually became Khorasan state in modern Iran, and "Greater Khorasan" is generally used to refer to the larger historical region.