Asha Greyjoy is Lord Balon's only surviving daughter and, in the eyes of most of the ironborn who've sailed with her, a better captain than either of the brothers who outrank her by sex alone. She commands the Black Wind, holds Deepwood Motte for her father's crown during the war the Citadel files under the ironborn's invasion of the North, and stands for the driftwood crown herself at the kingsmoot after Balon's death — arguing, more sensibly than most of the captains present cared to hear, that the islands cannot win a war against the entire mainland and ought to stop trying.
The kingsmoot chooses her uncle Euron instead. She leaves Old Wyk without the crown and, not long after, without much choice in her own course either: Deepwood Motte, the castle she once took and held for her father's crown, is retaken from her by Stannis Baratheon's northern campaign, and she is run down and captured with what remained of her crew in the retreat through the wolfswood — her fate from that point tied to his rather than her own house's.
Whatever becomes of her afterward remains, as the northern campaign's outcome does generally, unresolved in the surviving record. What is settled is her relationship with her brother Theon — genuine affection across a family not otherwise known for it, and very possibly the one loyalty in House Greyjoy that survives every kingsmoot, invasion, and betrayal the rest of the family manages to generate.