As the title states.. I have always seen in movies how the captain of a vessel has the authority to wed a couple. Where does this tradition begin and why? Do they have to be "ordained" in the same way a minister, etc would be, or does the authority come solely from his position on board the ship?
Well... the movies are lying to you. Ship Captains do not have the ability in most cases to solemnize a marriage. The exact origin of the myth that they can is murky, but probably lies in past where sea voyages would last months and the captain of a vessel was the ultimate authority. Also, even though they can not solemnize a marriage, Captains usually have to make an official entry in the ships log if a marriage takes place on board. This combined with marketing, and romance authors has led to this myth that they can.
Generally a ship has to follow the laws of the nation under whose flag it sails under. This includes laws about marriage. Both the US and British Navy have specific rules stating their Captains can not perform weddings. Confusing the matter further, in the US there are conflicting court cases about the legitimacy of weddings performed at sea. Some countries do allow and empower Captains to solemnize a marriage, but they are the exception rather than the rule.
Source : George Monger, Marriage Customs of the World.