The Throat of the World game guide below, is part of the full Elder Scrolls V: Skyrim walkthrough, but it refers strictly to the fourth main-quest.
The Throat of the World quest starts during your conversation with Esbern, after you discover how to defeat Alduin, in the third main-quest.
Your first objective is to visit the Greybeards who can offer additional information regarding the new shout that you have to learn.
The locations you have to visit in order to finish The Throat of the World are High Hrothgar, High Hrothgar Courtyard, Sky Haven Temple and Throat of the World.
Objective 1: Talk to Arngeir
After you reach Alduins Wall, Esbern will explain that in order to kill Alduin, you must learn a new shout.
To do this, you must travel to High Hrothgar and talk to the Greybeards.
While Esbern tries to find more clues engraved in Alduins Wall, exit the Sky Haven Temple, and go north until you reach a ruin.
Use Fast Travel and go to High Hrothgar, following the same path that you used during the fifth quest of the first act (The Way of the Voice).
Look for Arngeir and tell him that he must teach you the ancient Shout used to defeat Alduin.
You will notice that Arngeir will get angry; but after a dialogue with Einarth, he will apologize and tell you that the shout you have to learn is called Dragonrend.
Unfortunately, he cannot teach you this powerful Shout, but he knows who can. It is his master, Paarthurnax.
To get to Paarthurnax, you have to learn a new shout, called Clear Skies, because the path to Paarthurnax is surrounded by fog.
Objective 2: Learn the Clear Skies Shout
At this point, you have to follow Arngeir to the courtyard and learn the new shout, just as you did in the previous act.
You will learn the following words: Lok, Van and Koor, words that form the Clear Skies Shout.
Absorb the knowledge and then select the shout from your inventory.
Objective 3: Use the Clear Skies Shout to open the path
While standing near the firepit in the courtyard, go southeast and follow the steps up.
Make sure you have the Clear Skies Shout selected, and when you reach the fog, use it to clear the path in front of you.
Your objective will change.
Objective 4: Talk to Paarthurnax
Move forward slowly, and use the shout every time the fog surrounds you.
NOTE: The fog drains stamina, and it is important to know that if you try to advance blindly, you may die. Additionally, you may encounter several enemies, such as Ice Wraiths. If you run out of stamina, it is possible for them to defeat you.
When you reach the top of the mountain, you will hear a powerful voice. It is Paarthurnax, the dragon.
Tell him that you have to learn the Dragonrend Shout and he will explain that patience is required and that you have to prove that you are the true Dovahkiin.
Respond to his Thuum, and observe how Paarthurnax projects his fire breath to a nearby rock, revealing a new Word of Power.
Objective 5: Learn the Word of Power from Paarthurnax
Approach the stone and learn the Word of the Fire Breath Shout and then Paarthurnaxs knowledge, to receive your next objective.
Objective 6: Use your Fire Breath Shout on Paarthurnax
After you learn the new shout, Paarthurnax will ask you to greet him as a true dovah.
Open the inventory, select the Fire Breath Shout, and yell to Paarthurnax.
NOTE: Even if Paarthurnax will not get hostile when you use the shout on him, it is not a good idea to attack him with weapons.
Objective 7: Talk to Paarthurnax
Now, that you proved yourself, Paarthurnax wants to talk to you.
Approach, and when he asks what you want from him, repeat that you want to learn the Dragonrend Shout.
When Paarthurnax asks why you want to learn the Shout, tell him that you have to stop Alduin.
During the dialogue, Paarthurnax will ask you why he lives there, and you will not be able to answer correctly, but he will inform you that he is meditating in that spot, because this is where Alduin was defeated in the past.
You will also find out that Paarthurnax does not know the shout and that in order to defeat Alduin you have to learn more than the Thuum you are looking for.
In the past, the ancient Tongues defeated Alduin, using two elements: the Dragonrend Shout (the one you are searching for) and the Kel (Elder Scroll), which allowed them to generate a Tiid-Ahraan (Time-Wound) and send Alduin to the currents of time.
NOTE: At this point, you should ask Paarthurnax for additional information, especially about the Elder Scroll. If you found one, you will be able to travel in time and learn the Dragonrend Shout. If not, you will have to find the scroll, and start the next quest, called: Elder Knowledge.