Believe it or not, in order earn different Titles for your character in Sea of Thieves, you actually must complete a number of different steps. The first thing you must do is unlock titles, done either by Purchasing Promotional Ranks from the different Trading Companies or by Unlocking the Highest Grade of Commendations from the Trading Companies.
With every 5 Levels of Reputation your earn with the Trading Companies of Gold Hoarders, Merchants Alliance and Order of Souls, you will be able to purchase a brand new Promotional Rank. Not only will you be allowed to complete more complex Voyages with these higher ranks, but you will also unlock the Associated Title for each of them. Each Trading Company has 22 Different Titles for you to obtain.
However, purchasing and unlocking Titles from each of the Trading Company actually won't automatically equip you character with them. Instead, after obtaining the titles, you must go to any Clothing Store at any Outpost. It is then that you will find the merchant now sells the title you unlocked for free.
After purchasing your Titles, you can equip them to your character by going to a Vanity Chest (found outside of any Clothing Stores and in your Ship's Hold). (You can even swap between titles as you would so please.) Once your character has been equipped with a title, it will automatically display under their name.