Your Navy/Pirate rank determines what upgrades you have access to from a port, and what new ships you can buy. As well as the rewards you get from your faction leader.
Your rank goes up when you visit the faction promoter NPC at your faction base after recently acquiring enough faction fame to reach a new rank.
The fame requirements for each rank are as follows:
As you can see the potential is in place for a lot more ranks, the point to the rank system is that even if you reach Level 1000 you can still gain fame and reputation to increase your rank and get access to new content for vessels.