Thane of Whiterun is a rank given to the protagonist in Skyrim by the Jarl of Whiterun after completing the quest Dragon Rising.[1]
After receiving the title of Thane, you can buy a house in Whiterun, the Breezehome. This can be upgraded by talking with the steward of Whiterun. You are also rewarded with a random enchanted axe.
The Jarl will also give you a housecarl, Lydia
As the Thane of Whiterun, If you are caught committing a crime in the city, you can ask the city guards to ignore it. This will work only once.