Jarl Balgruuf will ask you to speak to his court wizard, Farengar Secret-Fire, and help him with his research into dragons. Farengar, who can be found in a room off of the main hall, will ask you to acquire an artifact called a Dragonstone from a ancient burial chamber known as Bleak Falls Barrow.
If you’ve already done the quest The Golden Claw, you should already have the Dragonstone in your inventory and you can just give it to him now. If you haven’t done that quest and you don’t have the item, then go to the town of Riverwood and speak to the general store owner and his sister to get The Golden Claw quest and do that at the same time as you do this one. If you have done that quest but didn’t collect the Dragonstone artifact when you had the chance, then just go straight back to the Bleak Falls Barrow dungeon itself and get it.
Our walkthrough of the quest The Golden Claw at that link will tell you everything you need to know to navigate this dungeon and retrieve the Dragonstone. When you have it, return the Dragonstone to Farengar to complete this quest.