This Quest is only available for Titans and unlocks your third sub-class, the Sunbreaker. Complete the very first mission of The Taken King's storyline, and Zavala will have the quest available.
The quest is a two-part mission that can only be played solo, and takes place in two different areas. Completing the first area will automatically send you to the second, with no ability to change destination, but the quest is short with few enemies.
Once you complete the second area of the quest, you will have the Sunbreaker sub-class, but you must still return to the tower and turn in the quest to Zavala, and only then will the achievement pop. Enjoy your new subclass and gamerscore!