There are for the moment three different story modes. The first of them are the battles against the pirates, which concludes in a final battle against Henry Morgan for your survival and for the Caribbean. The second story mode is similar to the first, but instead of pirates, you fight against the West Indian Company for the piracy survival and for the Caribbean.
The third story mode is recent in the game, and it's different from the other story modes. There story starts with Lady Tarrot, a friend of yours, which asks you to do several missions for her (sinking, capturing, escort and smuggling missions). In the course of the story, you meet with important characters, such as Captain Bloode and Angela, that also will ask you to do several missions for them. In the end, you will be forced to choose a side between this three characters.
In the end of the third story mode you will be able to make a choice between Lady Tarrot (peaceful end with Europe), Captain Bloode (War against Europe) and Angela (Hero in the Caribbean). Any of this choices will reward you with a goverment port of your choice to be you new base.