A Chance Arrangement

You have been informed that a man known as Brynjolf is the person to talk to if you are interested in The Thieves' Guild.

Speak to Brynjolf during daytime
Byrnjolf can be found in the marketplace in the centre of Riften between 8am and 8pm. Go to the marketplace between these two times and speak to him. He will tell you that he needs a job done after you express your interest in thievery to him. He will ask you to steal a ring from an Argonian called Madesi and plant it on another merchant named Brand-Shei while he distracts everyone.

Steal Madesi's ring
Madesi's Silver Ring can be found in Madesi's Strongbox under the counter of his stall. You would have to pick the lock to the sliding door and then to the strongbox. Be make sure you are hidden before picking the locks if you do not have the Quick Hands perk. The level of the locks is based on your character level. If you fail to pick the lock or have no lockpicks, you can pickpocket the key from Madesi. Once you have got the ring, head around to where Brand-Shei is.

Plant Madesi's ring
Brand-Shei will be sitting on a crate with his back to some barrels. Quicksave the game before you attempt this. To plant the ring on him, go behind the barrels and pickpocket him through a gap in the barrels. If he catches you, reload the game and try again. When the ring is planted, Bynjolf will cut his demonstration short and the people will begin returning to their positions.

Speak to Brynjolf
Just speak to Brynjolf and he will tell you to meet him at the Ragged Flagon in the Ratway.