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Walkthrough

When you return to the Throat of the World, Paarthurnax will recognize that you have the Elder Scroll, and he'll direct you to read it at the "time-wound" in the center of the summit. When you do so, you'll be taken back in time, where you'll witness the nord heroes Felldir, Gormlaith, and Hakon battling Alduin. The trio will immediately use the "Dragonrend" shout on him -- which will cause you to learn it -- but when they're not able to defeat Alduin even in his weakened state, Felldir will read the Elder Scroll and banish him to the future.

At the end of your vision, Alduin will finally make his appearance at the summit. He and Paarthurnax will fly around attacking each other, and you'll need to use the "Dragonrend" shout to make Alduin vulnerable and knock him to the ground. You'll only be able to damage Alduin while he's under the effect of "Dragonrend," so you might need to use the shout on him several times. For best results, you should keep shouting "Dragonrend" as quickly as possible, to keep Alduin vulnerable and on the ground. If you have trouble in the fight, then be sure to increase your fire / magic resistance, and also pick up the sword Dragonbane from Sky Haven Temple.

Note: Alduin will sometimes shout a meteor storm into existence. To counter this, use the "Clear Skies" shout.

When you defeat Alduin, he'll inform you that he can't be slain by you, and then he'll fly off. That will conclude the quest.