There are several kings of Persia named Darius. Darius 1, the Great, reigned from about 522 to 485 B.C. He continued the policy of Cyrus, of allowing the Jewish people to return to Israel. The Jews resumed the work of restoring the Temple in Jerusalem, in Darius' second year as king (Hag. 1:1; Zech. 1:1), and completed it in his sixth year (Ezra 6:15). Darius gave a great deal of help to the Jews in the rebuilding, including money and protection. In His final years as king, Darius led two battles against the Greeks, which were unsuccessful.
Darius the Mede was the successor of Belshazzar to the throne of Babylon. He was the one who made Daniel a governor (Dan. 6:1-2). It was also Darius the Mede who had Daniel thrown into the lions den, but later decreed that everybody in his kingdom "must tremble and fear before the God of Daniel" (Dan. 6:26-27).
Next person in the Bible: Dathan