This Albatross was first known as "The Scourge". Jack used the ship as his main offensive centre when he attacked Mos Le'Harmless in an effort to gain complete control over the Eastern Sea. When this failed, Jack renamed his ship "This Albatross", which he said would be a curse upon pirates forever.
Eventually, the pirates of Mos Le'Harmless rallied under Gentleman Mallard and attacked Jack in his hideout in the Cursed Archipelago. Eventually the battle became centred around This Albatross, where Jack finally jumped overboard, seemingly to his death.
Although the ship was likely destroyed after the battle, recent activity suggests the ship may have been revived along with its captain.