The Throat of the World

During this quest the Greybeards teach the Dragonborn all three words of the Clear Skies shout, Lok (Sky), Vah (Spring), and Koor (Summer) in order to reach the summit of the mountain. They also learn a word of the Fire Breath shout (the exact word depends on the words already known).

Background
After learning the Clear Skies shout, the Dragonborn is able to climb the Throat of the World to see Paarthurnax, leader of the legendary Greybeards.

Walkthrough
Find Arngeir in High Hrothgar and speak to him about defeating Alduin. He will become angry that the Blades are involved and if the incorrect dialog option is chosen he will refuse to help. At this point Einarth will speak in a voice that echoes around the building:

Arngeir will apologize and mention Einarth reminded him of his duty. He will teach the Clear Skies shout. From the courtyard, follow the path up the steps. The path will continue into some bad weather; use the Clear Skies Shout to clear the mist and continue to the Throat of the World to meet Paarthurnax. There is a frost troll and some ice wraiths along the way.

Before learning about Dragonrend, Paarthurnax will offer to teach a word of the Fire Breath shout. Paarthurnax will use the Fire Breath shout at the nearby word wall, allowing the Dragonborn to acquire a word of the shout.

While speaking with Paarthurnax, it is learned that no dragon can use Dragonrend, because it was the first Thu'um made by mortals and the concept of mortality is beyond their understanding. According to Arngeir, the Shout was made entirely with hatred for dragons - by those who suffered the slavery of Alduin. Learning the word would mean taking in the hatred that created them as well.

In addition, Paarthurnax allows the Dragonborn to Meditate on the Words of Power, adding bonus effects to shouts and the Dragonborn.

Paarthurnax reveals that Alduin was first defeated using the Dragonrend shout, but not without the aid of an Elder Scroll. After Dragonrend was used, the Elder Scroll was used on Alduin to cast him on the tides of time. However, Alduin was simply sent forward. Since that day, Paarthurnax has remained at the point where time was shattered. He stayed there, waiting thousands of years for Alduin to appear once more.

From here the next task is to find an Elder Scroll, though Paarthurnax does not know where it may be. He suggests talking to either Arngeir or Esbern, both of whom will direct the Dragonborn to the College of Winterhold.

Trivia

 * The quest is named after the Throat of the World, the tallest mountain in Tamriel and also the quest's setting.
 * The last of the dragon burial sites marked by the Map of Dragon Burials will be opened after this quest and Alduin can be seen resurrecting Viinturuth, if properly timed.
 * It is possible to reach the peak of the mountain by climbing around the storms blocking the path, use of a horse on the slope to the right of the main doors into High Hrothgar is the easiest way up the mountain. Paarthurnax will not be present if this method is used to circumvent the quest unlocking the mountain top.
 * There are Ebony Ore, Silver Ore, Orichalcum Ore and Malachite Ore veins to mine near the top of the mountain.
 * A dragon skeleton may be seen half-buried in the snow on the path to the peak, presumably killed during the battle between Alduin's dragon army and the Ancient Nordic Heroes that took place here at the end of the Merethic Era, when Alduin was banished from Mundus.
 * The Notched Pickaxe is also found at the previously mentioned Ebony Ore Vein.
 * Paarthurnax will call the Dragonborn "elf" if the Dragonborn is one of the elven races, "Argonian" if the player is Argonian, or "mortal" for the other races when the Dragonborn greets him with Fire Breath.
 * After being taught the first word of Fire Breath, if a dragon soul is spent on unlocking it instead of receiving Paarthurnax's knowledge of the word, the second word will be taught and unlocked.

Bugs

 * Sometimes, one can not learn the second Clear Skies word because it just disappears. The marker remains over the second word indefinitely. The quest will not be able to be completed from this point.
 * This may be caused by not following Arngeir outside immediately, or by entering conversation with him a second time before going to the courtyard.
 * Solution: Reload a save from before entering High Hrothgar to speak to Arngeir. (Reloading a save from within the courtyard after speaking with Arngeir does not seem to fix the problem.)
 * Solution: Follow Arngeir outside immediately after speaking with him, and watch him whisper the words onto the ground in frame to ensure all words stay lit.
 * If a companion were to stand between Paarthurnax and the word wall while he uses Fire Breath on the word wall the companion will attack Paarthurnax. Upon death he does not burn away and you do not absorb his soul and the mission cannot progress. Have the companion(s) wait before approaching the surrounding area of the word wall.
 * The Word of Power occasionally does not appear when Paarthurnax uses it on the wall.
 * When gaining the Clear Skies shout and trying to use it, it says "more dragon souls needed" fast travel somewhere else and it should be working.
 * Sometimes after learning two of the words to clear sky, the third will be visible but not be able to be learned. this bug will not interfere with completing the quest, use rank one or two to continue forth. Can be fixed by reloading an earlier save point.
 * Sometimes, when Dragonborn already knows the first two words to Fire Breath and has them unlocked with dragon souls, as Paarthurnax attempts to teach the third word of the shout, the word will not appear. Using Fire Breath on Paarthurnax will proceed to the next part of the quest, but unfortunately, rendering one incapable of learning the third word to Fire Breath.
 * The objective "Talk to Arngeir" may be impossible to complete due to Arngeir being stuck outside and refusing to move unless being attacked. When spoken to he only says "Come inside, the others will meet us there".
 * After unlocking the "Clear Skies" shout and reaching the "Throat of the World", Paarthurnax refuses to land or does not appear to be at the scene.

Der Hals der Welt (Quest) La garganta del mundo La Gola del Mondo Глотка мира (квест)