You are informed that Karliah wishes to speak with you and Bryjolf. Go and find her in the centre of The Ragged Flagon – Cistern.
After discussing the appropriate course of action regarding Mercer, she asks you both to meet her at the Standing Stone. If you have already been there it is called the Nightingale Hall and is directly south of Riften, very near to The Shadow Stone.
WARNING: It’s best to go there and meet her immediately after being asked to do so, as those who take too long to rendezvous with her at The Standing stone might encounter a bug in the game that causes her to prematurely depart the area, which if it happens makes this quest un-finishable.
Once you meet her at the Standing stone she discusses the Nightingales and takes you into the hall where she hopes that you and Brynjolf will become Nightingales.
Once inside the hall, activate the armor stone to receive the Nightingale Armor set. These are light armor pieces with some nice enchantments (they can be further improved with Void Salts). Equip them, then join Karliah and Brynjolf to take the oath to become a Nightingale. You can refuse the oath if you want, but to continue the Thieves Guild line of quests you must become a Nightingale, and are required to take the Oath in order to do that.
Join the others and stand on the vacant glyph symbol as marked. Nocturnal will speak and formally accept you as new Nightingales. After completing the Ceremony, Brynjolf will offer you the chance to become the leader of the Thieves Guild after the current situation is dealt with. This quest is now completed.