Skyrim:The Throat of the World

Quick Walkthrough
# Talk to Paarthurnax.
# Learn the Word of Power from Paarthurnax.
# Use your "Fire Breath" shout on Paarthurnax.
# Talk to Paarthurnax.

Detailed Walkthrough
Dragonrend
With the information about Alduin's defeat learned at Sky Haven Temple, travel to High Hrothgar to meet with the Greybeards.

When you arrive, you will first need to turn over the Horn of Jurgen Windcaller and learn the final Word for Unrelenting Force if you have not yet done so. Once you have, tell Arngeir that you need to learn the shout used to defeat Alduin. He will be quite shocked and ask where you learned about such a shout. While the Greybeards and the Blades do not have a history of cooperation, if you refuse to tell Arngeir where you came across the information, he will not help you. You can either tell him that you learned it from Alduin's Wall or that you were talking to the Blades. In either case, he will be strongly offended and will be reluctant to help. Depending on your response, Einarth may chastise Arngeir for his impertinence by saying, ''"Arngeir. Rek/Rok los Dovahkiin, Strundu'ul. Rek/Rok fen tinvaak Paarthurnax." meaning, "Arngeir. She/He is Dovahkiin, Stormcrown. She/He will speak (with) Paarthurnax"''.

Regardless, Arngeir eventually offers a hasty apology, saying the decision to help or not help you is not ultimately his. He tells you that the shout used to end Alduin's reign of terror is called Dragonrend. However, Arngeir cannot teach it to you because he does not know it and the origins have been lost. If asked, Arngeir will explain that the shout was created during the terrifying reign of the dragons and is a shout of pure evil, embodying all the cruelty, fear, and horror of the time. If you want to learn this shout, you must speak with Paarthurnax, the leader of their order. Further questioning reveals that Paarthurnax lives in seclusion at the peak of the Throat of the World and surpasses all of the Greybeards in the Way of the Voice.

Arngeir will offer to help you reach Paarthurnax and leads you out into the courtyard where you will be taught all three words of the shout Clear Skies. You are now ready to attempt the trek up to the summit of the Throat of the World. The path up is blocked by magical winds dealing severe frost damage, even with resist frost effects. Use the shout Arngeir gave you to clear away the damaging mists.
Paarthurnax
When you reach the summit, a dragon approaches. However, resist the urge to attack, as the dragon will land and address you: "Drem Yol Lok. Greetings, wunduniik. I am Paarthurnax. Who are you? What brings you to my strunmah ... my mountain?" After you explain your objective, Paarthurnax will respond: ''"Drem. Patience. There are formalities that must be observed, at the first meeting of two of the dov. By long tradition, the elder speaks first. Hear my Thu'um! Feel it in your bones. Match it, if you are Dovahkiin!" He will then shoot a bout of flame onto a wall and encourage you to participate in "tinvaak" (loosely translated, "conversation").

Inspect the word he left behind to learn a new Shout, and Paarthurnax will grant you his knowledge of it, saying "A gift, Dovahkiin. Yol. Understand Fire as the dov do. Now, show me what you can do. Greet me not as mortal, but as dovah!"'' Use this new shout on Paarthurnax. He will agree with the opinions voiced by Greybeards that "the dragonblood runs strong in you", even if you are a "joor" (mortal), and offer to answer your questions.

After the formalities, ask Paarthurnax about Dragonrend. He will admit that he was expecting this, as "Alduin and Dovahkiin return together". Though he doesn't know the shout, he believes it could be learned by returning to the beginning of the "Tiid-Ahraan", the Time-Wound on the Throat of the World. An Elder Scroll, or "Kel" would be necessary to activate the portal and allow you to view the first Tongues using the Thu'um to defeat Alduin. As the quest ends, the dragon admits he has no idea where to find an Elder Scroll. Your quest log suggests you talk to Arngeir or Esbern for more information.

Notes
If you have already acquired an Elder Scroll by completing the Discerning the Transmundane quest, Paarthurnax urges you to the Time-Wound, and the quest Alduin's Bane will begin immediately.
You can use the Become Ethereal shout to stroll through the mists except for the first cloud; passage through it requires Clear Skies.
If you unlock the word yourself at just the right time, before Paarthurnax gives you his "understanding", he will teach you the second word for Fire Breath instead of the first (or the third if you started out knowing the second). 
Paarthurnax will say that he is teaching you Yol, no matter which Word the wall actually teaches you.
It is possible to climb to the very top without going through the temple and the barriers.
Paarthurnax and your follower may become hostile towards each other. In addition to the possibility of harming your follower, this may also prevent you from talking to Paarthurnax. This normally can be avoided if you have your weapons sheathed prior to entering the area.
Viinturuth will be resurrected by Alduin once this quest is completed, at the mound northwest of Anga's Mill and near Lake Yorgrim.

Choose the optional dialogue and say you want to learn about: 
:
:These increase the respective shout capabilities. You can only have one active at a time.

Bugs
 
Reloading a save from before talking to Arngeir then following him outside immediately will usually fix the problem. Sometimes it is necessary to watch him whisper to words on the ground to ensure they stay lit. 
 
 
 
  In the Console, enter   to unlock the first word. If you already have one or two words of Fire Breath, use   or  , respectively.

Quest Stages