When your Pirate has a quest, you will notice a yellow arrow with text near the bottom of the screen. The text and the yellow arrow are called the Quest helper, which reflects information pertaining to the Quest you have selected.
To reach the next step in your Pirate's quest, just run in the direction the arrow is pointing and complete what the text below asks you to do.
Mystery Quest Arrow/Explore Goal
Sometimes the point of a quest is to find hidden items in the Pirate101 game world, and for these quests the standard quest arrow is replaced with a mystery arrow! That way you can use your skills to find the hidden item in order to complete the quest.