Over the course of Persian history, there have been several kings named Darius. The most significant are:
Darius the Great, who lived until 486 B.C., and is also known as Darius I. He is mainly known for his ill-fated attempt to militarily punish the Greek city-states, especially Athens, because a couple of them had helped Aristagoras, who was a leader in the Ionian colony city of Miletus, when he rebelled against the Persians who had annexed most of Asia Minor and were demanding tribute payments from these Greek colony cities. This attempt by Darius to punish Athens was defeated at the famous Battle of Marathon in 490 B.C.
Darius III fought against Alexander the Great, and was assassinated by one of his own officials in 330 B.C.
Darius the Mede is recorded by Hebrew historians as conquering Babylon. Due to obscurities in translation and transliteration, this reference is somewhat unclear; it could refer to Darius the Great, or it could be a way of referring to Cyrus; it could allude to one of several kings of the Medes; it could also indicate Ugbaru-Gubaru, who was a military leader of the Medes. Yes, that is a real name.
There are a number of other kings and leaders named Darius in Persian history, but these are the most important.