Cyrus the Great, or Cyrus II, was so great because he was the founder of the Persian Empire. He was born somewhere between 590 B.C. and 580 B.C.

CYRUS'S FAMILY
The historians Herodotus and Xenophon are telling us that Cyrus' father was the Persian ruler Cambyses I and Cyrus' mother was Mandane, the daughter of Astyages, king of the Medes.

CYRUS'S RULE
Cyrus II ruled from 559 B.C. until his death in 529 B.C.

At that time, the Babylonians were the ruling world power. In 539 B.C., twenty years after becoming king, Cyrus the Great seriously kicked Babylonian rears.
 

WHO FOLLOWED CYRUS THE GREAT TO THE THRONE?

Cyrus II the Great
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Cambyses II (son of Cyrus II the Great)
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Smerdis (who may or may not have been the real Smerdis, son of Cyrus II the Great)
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Darius I the Great
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Xerxes I (son of Darius I the Great)

In detail:
Among others, Cyrus had three children:
His son Smerdis (or Bardiya), his son Cambyses II, and his daughter Atossa.

When Cyrus died, Cyrus's son Cambyses II became king. Cambyses II was pretty useless in character if you want to trust historian Herodotus. Herodotus said that Cambyses II committed many atrocities while conquering Egypt.

After killing his brother Smerdis, Cambyses II ruled Persia for seven years and died. Then, the magician Gaumata claimed to be Cambyses's brother Smerdis and ruled for a while.

Darius I killed Gaumata, the usurper, married Atossa and ruled himself. At least that's how Darius I told the story. Some historians think, Darius I might have actually killed the real Smerdis.
 

Go here for a complete list of the Achaemenian rulers

By the way, Darius wrote all this down on the Behistun Rock.