Talk to Brynjolf in town who will ask you to help him teach someone a lesson. Brynjolf will create a distraction and you will have to go over to Madesi's stall and steal the ring out of the lock box in it. You will have to pick the lock but watch out for the guards on the wall, they may spot you. Once you have the ring, sneak up behind Brand-Shei and reverse pickpocket the ring into his inventory. Go back and talk to Brynjolf to be invited into the guild.
As an alternative, you can fail this quest in any number of ways, simply by waiting for Brynjolf's diversion to run down, by going into a building and returning to Brynjolf, or simply by being caught while stealing the ring or pickpocketing it. If you fail, it's ok because you will still be invited to join.
