I have seen this armor in Skyrim around the internet, and I have no idea how to get it. I think it may have something to do with Thieves Guild, but I don't know any more. How do I get this armor?
From the Skyrim wiki:
The full set is given to the Dragonborn by Karliah before performing the initiation ritual of the Nightingales during the quest "Trinity Restored" for the Thieves Guild. All pieces are obtained when activating the Armor Stone, emblazoned with the Nightingales symbol, inside Nightingale Hall.
unlock "perfect touch" perk in pickpocketing skill tree. Join the thieves guild until you reach "trinity restored" quest line. After finishing the quest, sneak and pickpocket karliah and get all of her nightingale equipments.
First of all, you need to do some thief questline quests, you'll meet Karliah in some of these quests, make sure to do they and she'll make you become a Nightingale!