The ancient Persian Empire included areas far beyond modern Persia, now called Iran. At times the Persians controlled much of the Middle East — they were a main antagonist of Ancient Greece a few centuries BCE, and later the Roman Empire. They ruled Egypt at one point — as well as much of the Caucasus and Central Asia and parts of what are now Pakistan and India. At the height of its power, more than 40 per cent of the world's population were Persian subjects, a higher ratio than any other empire ever.
In Western canon, the Persian Empire was famous for the liberation of Jews from Babylon, as well as the Persian Wars with Ancient Greece.
Perhaps the longest on-again-off-again war in recorded history was between the Persian and Roman empire beginning with an ill-fated expedition under Crassus in the first century BCE, enduring past the replacement of Parthian with Sassanian Persia and the fall of the Western Roman Empire and only coming to a close with the defeat of Persia (much weakened by a then recent fight with Rome) at the hands of Muslim invaders.
Persia has been conquered three times: by Alexander the Great in the 4th century BCE, by Arabs during the great expansion of Islam in the 7th century CE, and by the Mongol Empire in the 13th. Every time, it has risen again to create another Persian Empire. The Sunni-Shia split coincides with the Persian-Arab cultural split to a large extent (historically Persian lands being largely Shia) and some argue that the cultural differences rather than religious or theological ones are the main reason for the sectarian violence in modern times.
Persia had a huge influence on Central Asia, much of which they ruled for centuries. Marco Polo, for example, describes cities like Bukhara and Balkh as Persian. Even today, one dialect of Persian is the main language of Tajikistan and another is widely used in Afghanistan. They also had a tremendous influence on South Asia which was repeatedly invaded by Persian speakers, from Darius taking Gandhara in the 6th century BCE to the Mughal Empire which ruled much of the subcontinent from the 16th century CE into the 19th.
The modern game of chess is believed to have originated in the Persian game shatranj, which also gave rise to other chess variants in parts of the world such as Chinese Xiangqi, Japanese Shogi, and the Korea and Thai variants. While shatranj originated in the India game chaturanga, it was the Persian version that spread to other parts of the world and gave rise to all the modern-day national and international chess variants. Some chess terms also go back to Persian terms "checkmate" comes from "shah mat" or "the King is dead", and the "rook" takes its name from the Persian word "rukh", meaning "chariot".
Persia also had a strong influence on religion in various areas; for one thing, the ancient Persian faith Zoroastrianism still exists, mainly in India and Iran. As for Christianity, the empire supported the Church of the East, apparently mainly for political reasons; the Persians did not want a church with strong ties to either Byzantium or Rome becoming too influential in their territory.
That church never accepted the condemnation of Nestorius as a heretic by the western bishops. It sent missionaries east along the Silk Road to spread its Nestorian version of the Gospel. They reached China and Korea by the 7th century, hundreds of years ahead of other Christians. Later, Persia sent Muslim missionaries along the same routes.
Old Persian was only one of many languages spoken and used officially by the First Persian Empire, the Achaemenid, along with Babylonian, Elamite, Aramaic and even Greek. This multilingual approach to governance continued during the course of the Parthian and early Sasanian Empires. By the late Sasanian however, Middle Persian had emerged as the prestige and dominant language in the Greater Iranian sphere, which over the centuries evolved into Modern Persian and kept its dominance to the modern day.
The main dialects of modern Persian are Farsi in Iran, Tajik in Tajikistan and Dari in Afghanistan. They are different enough to create difficulties, but not to completely prevent understanding.
At its peak, around 500 BCE, the empire was enormous. These areas retained Persian culture for centuries:
Around 330 BCE Alexander the Great conquered all of this empire. The Seleucid Empire, founded by one of Alexander's generals, ruled Persia and some nearby areas for a few hundred years after that
See Iran#Cities for other modern towns there.
Related Wikipedia article: History of Iran