Possibly the most famous of the Barbary pirates were Oruç Reis and his brothers Hızır and Ishak, known as the Barbarossa or Redbeard brothers. Born Ottoman Turks, they began their careers as corsairs in the service of the governor of Anatalya. After some success they moved to the Western Mediterranean in 1504, basing themselves in Tunisia in return for sharing their booty with the Sultan of Tunis. As well as general piracy, they also smuggled Muslims out of newly reconquered Spain to North Africa.
Later they captured Algiers from the Spanish and Oruç usurped the throne of the Sultan of Algiers. Under increasing pressure from the Spanish, Algiers joined the Ottoman Empire in 1517, and Oruç was appointed Governor (or Bey).
Oruç and Ishak were killed while fighting the Spaniards and their Bedouin allies at Tlemcen, leaving Hızır, better known as Hayreddin Barbarossa, as Bey of Algiers.
But why are they known as the Red Beards? The picture on his Wikipedia page shows that Hızır's beard is black, as might be expected from someone half-Greek and half-Turkish. What's going on?
Sadly, the reason is that Oruç was called Baba Oruç (Father Aruj) by the Muslims he helped out of Spain. This sounded like Barbarossa to the Sicilians and Genoans who were amongst the brother's enemies. So, just as dying your beard blue doesn't make you a pirate, neither does dying it red. And Pink is right out.
 On Lesbos. Stop smirking.
 Look, the whole thing is terribly complicated, what with Christian Spaniards appointing Muslim puppet governors, despite having kicked all the Muslims out of Spain.