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When you talk to Brynjolf in the Riften market, he'll ask you if you want to help him with an "errand." If you say yes then you'll trigger the quest.
Brynjolf will tell you that he's been hired to put a merchant named Brand-Shei out of business, but that he needs "an extra pair of hands" to get the job done. He'll then tell you his scheme: while he's creating a distraction, you'll need to steal Madesi's Silver Ring from Madesi's stall and place it in Brand-Shei's inventory.
To put the plan into action, you'll need to talk to Brynjolf in the market during the day. This will cause him to start advertising an "amazing" elixir, which will draw everybody over to him, so they'll be looking at him rather than at you. You'll then need to pick open the sliding door of Madesi's stall, which will give you access to Madesi's strongbox, and then pick open the strongbox to acquire
Madesi's Silver Ring
. Finally, you'll need to sneak behind Brand-Shei and pickpocket him so you can place the ring in his inventory.
Note: Brynjolf's performance will last as long as necessary for you to complete the above, so feel free to do some extra stealing while you're there.
Atferwards, when you talk to Brynjolf, he'll pay you 100 gold, and he'll invite you to join the Thieves Guild. This will trigger the next quest,
Taking Care of Business