This card is a reference to The Rime of the Ancient Mariner, an 18th century poem by Samuel Taylor Coleridge. In the story, an albatross, typically seen as good fortune due to flying on favorable winds, is shot and killed for meat by a sailor. As a result, the crew suffers nothing but bad winds and rough seas. Thus, they tie the corpse of the albatross around the offending sailor's neck for bringing bad luck onto them.
The Albatross tokens created by Bad Luck Albatross originally had no class defined in the game's code (neutral cards are internally defined as their own class), so they were not treated like neutral cards. This meant that Lightforged Zealot and Lightforged Crusader could still trigger their Battlecries even with Albatrosses in the deck and Tess Greymane could resummon any played Albatrosses. This bug was later fixed.