In the books
In the A Song of Ice and Fire novels, Blacktyde does not have a very distinguished history. However, at the beginning of the narrative its current lord is Baelor Blacktyde, who spent eight years as a ward on the mainland after the Greyjoy Rebellion, much as Theon Greyjoy was sent as a hostage to Winterfell. Unlike Theon, Baelor Blacktyde spent his years as a ward in Oldtown in the Reach, one of the major centers of Andal culture. When Baelor returned (when he came of age, seven years ago), he was an ardent convert to the Faith of the Seven. Due to Baelor Blacktyde's household, Blacktyde is the only one of the Iron Islands with any minority to speak of who follow the Faith of the Seven instead of the Drowned God.
During the Kingsmoot, Asha (called Yara in the show) Greyjoy's main base of support was Harlaw island, ruled by her maternal uncle, Rodrik "the Reader" Harlaw. However, Baelor Blacktyde also openly sided with her - due to the loss of his father in the last rebellion he was more receptive to her warnings about the losses they would take if they kept antagonizing the mainland.
After Euron wins the Kingsmoot, Asha manages to slip away and escape, and many of the lords who voted for her grudgingly don't challenge Euron as the new king. The exception was Baelor Blacktyde, who adamantly refused to accept Euron as his king, announcing "Balon was mad, Aeron is madder, and Euron is maddest of them all". Euron promptly had him executed, then had his corpse cut up into seven pieces, in mock honor of his faith.