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Outside the Hall of Valor, you'll find the heroes Gormlaith, Hakon, and Felldir just past the Whalebone Bridge. When you reach them, they'll suggest that the four of you shout Clear Skies together to remove the mist. The heroes will then wait for your signal, and they'll shout when you shout. The first two times you try this, Alduin will simply shout the mist back into place, but the third time he'll lose his patience, and he'll attack you.
The battle against Alduin is roughly the same as battles against other dragons, except for three differences: you'll have three heroes (plus assorted other spirits) helping you out, Alduin will frequently call down meteor storms to damage you, and you won't be able to damage Alduin unless he's affected (that is, glowing blue) from the Dragonrend shout. So hit Alduin with Dragonrend as much as you can, damage him when he's affected, and protect yourself when he's not. The heroes will do a good job of distracting him, so if you take too much damage, you should be able to get out of the way and heal yourself. When Alduin dies, he won't leave behind a corpse to loot, and you won't be able to absorb his soul.
After the battle, Tsun will congratulate you on your deed. "They will sing of this battle in Shor's hall forever." To exit Sovngarde and return to Skyrim (where you'll be able to complete more side quests if you want), you'll just need to talk to Tsun and ask him to shout you back. He'll do this, and he'll also teach you the shout Call of Valor, which will allow you to summon a hero from Sovngarde to fight with you.
Back in Skyrim, you'll appear at the Throat of the World, where several dragons will acknowledge your victory over Alduin before departing. Paarthurnax (if he's still alive) will then talk to you. He'll mourn Alduin's loss, but he'll conjecture that most of the remaining dragons will start following his lead rather than continuing to menace Skyrim. Don't worry though. Random dragons will still attack you from time to time, and so you'll be able to continue gathering resources for crafting and souls for Words of Power.
Congratulations on completing the main questline in Skyrim!