Soon after you return the Dragonstone to Farengar, a scout will approach and inform him that a dragon has been sighted outside of Whiterun. The scout asks you to go and speak to Jarl Balgruuf. The Jarl will ask you to accompany his Housecarl Irileth to the walls of Whiterun and set out with her and some of his other troops on a mission to kill this dragon before it attacks the city.
Rendezvous with Irileth at the Western Gate where she is making a speech to rally her soldiers for the coming battle. When she’s finished, follow her and the other soldiers out of the city and towards the western watchtower.
You will arrive to find that the watchtower has been reduced to a smoldering ruin; the dragon has already struck. After a short talk, Irileth will give orders to search for survivors. On the north side of the tower you will find a surviving Whiterun Guard. Shortly after talking to him, the dragon will come flying back over the mountain.
The dragon, named Mirmulnir, will follow a regular pattern of attack: it will land, fight, take off an fly around some more after taking a certain amount of damage, before eventually landing again to fight some more and taking off again when damaged some more and so on…
When on the ground, the dragon has two attacks: it will breathe fire and it will strike. While you are facing the Dragon at a short distance, it will breathe fire at you while you remain in its line of sight. When you are up close to it, it will attack with its claws or teeth to inflict massive damage or send you flying through the air. If you try to approach it from behind, it will beat its tail into the ground. After losing roughly a quarter of its health, it will fly into the air and circle around a few times before landing on the ground again.
The best tactic is to approach the dragon from the side while it is occupied with the other soldiers approaching it from the front and strike the side of its body relentlessly with weapons or spells until it flies into the air again. Pay attention to what the other soldiers are doing because if enough of them die, the dragon will start to focus its attention entirely onto you and you’ll find yourself always facing it head on when it’s on the ground.
When it finally falls, the Dragon will disintegrate and release its soul in a blast of energy, allowing for the use of the Dragon Shout (Unrelenting Force) that you learned by reading the inscription on the word wall during the Bleak Falls Barrow / Golden Claw quest. After using the shout, a nearby guard will approach you in awe and exclaim that you are a Dragonborn. Some of the other guards are dubious of this statement and begin to argue with him. Irileth discards it as a myth and walks off to investigate the wreck, after telling you to report back to the Jarl.
On the way back to Whiterun, you will likely hear a thundering voice from somewhere off in the sky boom out “DOVAHKIIN”, shaking the very ground in the process. Make your way back to Dragonsreach and go and see Jarl Balgruuf.
After you recount the events at the watchtower, and he learns of you being dragonborn, he informs you that the bellowing shout you heard on your way back to the city was the order of monks known as the Greybeards summoning you to their mountain temple of High Hrothgar. He says that they can teach you to use the gifts of the dragonborn. The Jarl advises you to travel to High Hrothgar and speak with the Greybeards immediately, as the summoning is a great honor.
For your assistance in slaying the dragon, the Jarl will make you a Thane of Whiterun, and assign you your own personal Housecarl; the Nord warrior Lydia. He also awards you the Axe of Whiterun as a symbol of office.
From now on you also have the right to own property in Whiterun. Speak to the Jarl’s courtier if you want to buy a house in the city at any point in the future.