Vex will explain that it is one of the Stones of Barenziah. She will also say that it is not worth much by itself, and that there are 24 of these stones. She asks the Dragonborn to come back with the full set. For locations of all gems, see here.
If the Dragonborn is already a member of the Thieves Guild and has shown the gem to Vex, and then talks to Maul in Riften about the stone, he will suggest becoming a member of the Thieves Guild, adding a quest objective to join the Thieves Guild. While this objective cannot be cleared, it will not prevent the quest from being completed.
If the Dragonborn already possesses all 24 gems when talking to Vex for the first time, the game will not recognize them, and the quest will be incompletable. Collect 23 or fewer gems before talking to her to prevent this bug.
This bug has reportedly been fixed by patch 1.9.
Sometimes the quest objective will count extra stones that have not been collected (i.e., saying 10/24 Stones have been collected when the Dragonborn only has 8). The quest can be completed when the objective says 24/24, even if some stones have not been collected yet.
If you start the Dark Brotherhood quests and don't pick up the stone after the sanctuary burns, it is no longer available which stops the quest indefinitely.
PC Fixes for this quest if it becomes bugged or incompletable. Note that these may not update the journal, but will unlock the appropriate responses from Vex. On the console type the following commands:
To initiate the search for the crown (if stones are missing)
set TGCrownGemsFound to 24
set TGCrownGemsTotal to 24
setstage TGCrown 30