The pirate ship commanded by Edward Kenway in Assassin's Creed IV is the Jackdaw.
Edward acquires the vessel early in Black Flag and uses it to travel across the Caribbean. Players can upgrade the Jackdaw's hull, weapons, cargo capacity, and crew, making ship improvement a central part of the game's progression.
The Jackdaw supports naval battles, boarding actions, treasure hunting, and exploration. Its name comes from the jackdaw, a member of the crow family, and fits the game's pirate imagery.
Other Assassin's Creed ships are easy to confuse with it. The Morrigan belongs to Shay Cormac in Rogue, while the Aquila is associated with Connor in Assassin's Creed III. Neither is Edward's vessel.