To START this quest, you need to have found an "Unusual Gem" during your travels. There are quite a few of them, and chances are that unless you went to the Thieves Guild asap you've run across one.
Once you are part of the Thieves Guild proper, you can show the Gem to Vex, to ask her what the hell is it?! She will tell you that they are the "Stones of Barenziah", and that alone they are worthless... you need TWENTY FOUR (24) of them to make the SET complete. Once she tells you this, the quest in ON in full force. Our goal? Collect the rest of the stones!
Whiterun Catacombs: In the first room, go down the left stairs and it will be to your left, next to a skeleton. The catacombs are located within the Hall of the Dead.
Whiterun Dragonsreach: the Jarl's bedroom.
Whiterun Jorrvaskr's living quarters: on the bookshelf in Kodlak's room.
Solitude Blue Palace: Jarl's bedroom.
Solitude Proudspire Manor: Buy the house or pickpocket the Key to Solitude Fletcher.
Riften Mistveil Keep: Jarl's bedroom.
Windhelm Palace of the Kings Upstairs: Take the first door to the left upon entering, then find Wuunferth's room at the end of the hall.
Windhelm House of Clan Shatter-Shield: on a bookshelf in the room to the left of the top of the stairs.
Markarth The Treasury House: In an end table in Thonar's bedroom, which is down the hall on the left from when you enter the front door. You may need to pick an apprentice-level door, but no one reacts.
Markarth Understone Keep's Dwemer Museum: On a table in the first room to the left. Note: Do not go into the museum after completing the quest Hard Answers until you have finished the quest Darkness Returns. See the Hard Answers quest page for details.
College of Winterhold Arch-Mage's Quarters: Access may be gained at the end of the quest Under Saarthal .
Dead Crone Rock: On the stone altar opposite a Dragon Word Wall.
Black-Briar Lodge East of Riften: The gem is in the top floor bedroom, next to the bed.
Ansilvund: Northeast of Shor's Stone and due north of Riften.
Stony Creek Cave: Just north of the Ansilvund Burial Chambers, on the table in the last room.
Rannveig's: Fast After falling down the trapdoor, in front of the chest and Dragon Word Wall.
Fellglow Keep Northeast of Whiterun: Head straight up the stairs and to the right.
Dainty Sload Ship: Along the coast northeast of Solitude, in the last room on the table.
Sunderstone Gorge: On the table with two bodies in front of the Dragon Word Wall.
Yngvild (The island northeast of Dawnstar): The gem is in the bedroom behind the throne in Yngvild Throne room.
Hob's Fall Cave (On the northern coast, directly between Dawnstar and Winterhold): The gem is near the alchemy set.
Reeking Cave (or Thalmor Embassy after patch 1.4): inside Reeking Cave and next to a corpse in the back. Prior to patch 1.4, the stone is located in the Thalmor Embassy in Elenwen's Solar on the second floor.
Pinewatch Bandit's Sanctuary: In a house west of Helgen, there is a secret hideout accessible from the basement. The gem is found in a locked room in the Bandit's Sanctuary.
Dark Brotherhood Sanctuary: On the dresser in Astrid's room. It is still accessible after the Death Incarnate quest.