The term "black ship" in this story is used to refer to any of the ships that was used (usually only once a year) to ferry money and some goods from Asia back to Europe.
During the time in which this story is set, Spain was the only European country with a real presence in East Asia. Therefore it had a monopoly on trade there. Each year, a ship would go from Asia back to Spain carrying the money that had been made in that year's trading. The ship belonged to the Spanish government/crown.
Because the ship had such huge amounts of money, it made a very tempting target for people like Blackthorne and losing it would be a giant blow to Spain.