Its called Stones of Barenziah! Not Stone. Its known as Unusual Gem. This prompts a Miscellaneous Quest to be added to the Journal, saying that the player must appraise the gem before using it. Gems can be appraised by a few people; two of which are Maul in Riften and Vex, part of the Thieves Guild, also in Riften.
Upon getting it appraised, the player learns that 24 exist in total and are only valuable as a whole set. Most, if not all, are found in the extra jobs for the Thieves Guild. The reward for collecting all of these items is the thieves guild perk, Prowler’s Profit. (Chance of finding additional gems in chests.)
Now if you do not want to know where to look do not read below.Spoiler:LocationsEdit
Solitude, Proudspire Manor: master bedroom (you have to buy the property first: To do this you need to win the Jarl's acceptance by doing sidequests given to you by stewards and have 25000 gold)
Solitude, Blue Palace: on a shelf in Jarl Elisif's quarters
Thalmor Embassy: In Elenwen's Solar, in a bedroom (during or after the main quest "Diplomatic Immunity"
Dainty Sload: on a table in the Captain's Quarters (ship is moored near Solitude)
College of Winterhold: shelf in Arch-Mage's quarters
Yngvild: chamber behind the throne in throne room
Hob's Fall Cave - necromancer's sleeping area
Markarth, Treasury House: next to bed in the master bedroom
Markarth, Understone Keep: on a table in a locked side room of the Dwemer Museum
Dead Crone Rock: on a makeshift altar in front of the Word Wall at the top of the final tower
Whiterun, Jorrvaskr (Companions): In Kodlak's bedroom
Whiterun, Catacombs inside of the Hall of the Dead: In one of the wall crypts near a skeleton
Whiterun, Dragonsreach: in the Jarl's bedroom area
Rannveig's Fast: on a table near the watery prison
Fellglow Keep: on a counter in the workroom at the top of the front foyer
Windhelm, House of Clan Shattershield: Upstairs first bedroom on the left.
Windhelm, Palace of Kings: on a table in Wuunferth the Unliving's quarters (the first door on the left side)
Stony Creek Cave - in the Bandit Wizard's cavern
Ansilvund: near Fjori's ghost in the burial chambers
Sunderstone Gorge: on the alter in front of the word wall
Dark Brotherhood Sanctuary - on the dresser in Astrid's room (must be done during or after DB quests "With Friends Like These" and / or "Destroy the Dark Brotherhood"
Pinewatch: Move through Pinewatch until you get to the Pinewatch Bandit Sanctuary. Once in the Bandit Sanctuary defeat 3 bandits in the first room before moving up the ramp on the right. In the next hallway defeat 1 bandit before moving up a flight of stairs. In the next room you will find empty draugh tombes, directly to your right you will see a locked door. Lockpick this door and inside you will find the stone on the top shelf of the right bookcase.
Riften, Mistveil Keep: in the Jarl's chambers
Riften, Black Briar Lodge: in the upstairs master bedroom
Recovering the Crown of BarenziahEdit
After handing in the 24 Stones of Barenziah, the Dragonborn is asked to recover the Crown of Barenziah from Tolvald's Cave. Vex revealed that the the Crown of Barenziah is actually a paragon for the Thieves Guild . Recovering the crown, she should be able to restore the paragon to its full strength.
RewardEdit
After you have handed the Crown of Barenziah and the Stones of Barenziah to Vex and finished completed No Stone Unturned. She rewards the Dragoborn with the "Prowler's Profie". Afterward, the fully restored Crown of Barenziah sits on the mannequin bust in the Thieves Guild.