Founded by Cyrus II the Great (Kūruš) in the area of Babylonia (in southwest Iran and Mesopotamia), the Persian Achaemenid dynasty ran from about 560-330 B.C., when Alexander the Great defeated Darius III of Persia. Achaemenes was a semi-legendary king and ancestor of Cyrus, but he wasn't the start of the imperial dynasty. That was left to Cyrus when he overcame the Medes and united the related Persians with them. He then conquered further territory in Asia, including Babylonia, which welcomed him. His descendants expanded the Achaemenid Empire further.
The Achaemenid Empire was divided into twenty satrapies which allowed relative regional autonomy to the areas. At its height, under Darius the Great, the Achaemenid empire stretched from the Indus River to (late period) Egypt. By the end of Darius the Great's rule there were royal roads extending from Susa to Sardis, complicated irrigation systems, standardized coinage and more, while allowing the conquered peoples to maintain their languages, religions, and other elements of their cultures.
Although the official language of the Achaemenid Empire was Old Persian, the most common language was Aramaic. Most of the ancient references to the Achaemenids come from Greek sources (Herodotus, Aeschylus, Xenophon, and Ctesias).
Reference: "Seals and the Elite at Persepolis: Some Observations on Early Achaemenid Persian Art," by M. B. Garrison; Ars Orientalis, (1991), pp. 1-29.