The Horn of Jurgen Windcaller is unlocked after completing the quest "The Way of the Voice" and passing the Dragonborn tests in High Hrothgar.

Open your quest journal:
The Horn of Jurgen Windcaller

After demonstrating my ability to learn Words of Power and to Shout, the Greybeard Arngeir has sent me on a final trial - retrieving the Horn of Jurgen Windcaller from his tomb in Ustengrav, a burial mound in the marshes of Hjaalmarch

Retrieve the Horn of Jurgen Windcaller

Assuming you do not want to take the 7,000 steps down, you should fast travel to Morthal which is the closes town to Ustengrav. If you haven't been to Morthal yet, fast travel to Whiterun and from their take the horse carriage to get to Morthal.

From Morthal, travel Northeast to get to Ustengrav. Defeat the bandits guarding the circular entrance, and head down the spiral stairs. Enter through the door. Kill the enemies in the first room, then take the tunnel to the Northeast. Defeat some mages and Draugr you encounter. Head east, then south. In a small chamber to the west of the corridor you can pull a chain to open a section in the wall and find a room with a chest. Go back to the junction and then go south. Go east and climb some stairs, then west and go across the bridge. Head south and enter the door to the Ustengrav Depths.

Keep going and reach a big grotto. Go south, then east, cross a bridge, and arrive at some sort of dining room. Go past it to the south, defeating the draugr along the way. Take the stairs around, and cross the bridge. Fight the Draugr that emerge from two coffins. Go NW and decsend down to enter the Grotto again. Kill the skeleton sitting on the throne to the North.

Cross the bridge to the Southeast and deal with the Draugr archer.

To the east are three closed gates, and near it are three stones that illuminate light as you approach them, and each of them opens one of the gates for a few seconds. This is a PUZZLE

Stand behind the stones, to the center, and facing the gate, as shown in the picture below:

Ustengrav Puzzle Solution

Equip your shout: WHIRLWIND SPRINT

Start running in a straight line towards the door (without using the shout yet). As you pass each stone, a gate will open. As soon as you pass the third stone and all gates are open, IMMEDIATELY use the shout twice as you continue running to sprint through the open gates before they close.

Climb the steps and continue east. Watch for fire traps. Defeat the frostbite spiders in the next room and keep going east towards the gate. Pull the chain to open it.

Enter a big chamber. Four dragon statues will emerge from the water. Cross the bridge to find that the horn is gone from its pedestal, and instead someone left a mysterious note! Read it. It says:

"Dragonborn. I need to speak to you. Urgently. Rent the attic room at the Sleeping Giant Inn in Riverwood, and I'll meet you. --A friend"

You will have FAILED the objective of retieving the horn, but that's NORMAL so don't reload your last save. See the quest log:

"Unfortunately, when I reached the tomb, I found that someone had replaced the Horn of Jurgen Windcaller with a note asking me to meet them in the Sleeping Giant Inn in Riverwood."

Objective - Meet with whoever took the Horn.

Go through the door behind the pedestal to enter a treasure room, then Follow the tunnel to the north and take the door to Ustengrav. Activate a lever at the end of the corridor. Then go west to exit the tomb.

Fast travel to Riverwood. Go to the Sleeping Giant Inn. Talk to Delphine, the Innkeeper. Tell her you'd like to rent the attic room. She will tell you they don't have an attic room but you can have the room to the left.

Enter the room and close the door behind you. Delphine will come in after you. She will give you the horn and ask you to follow her. She will lead you to her room and down a secret passage behind her wardrobe. Listen to what she has to say. She needs you to help her stop whatever is bringing dragons back to life. She wants to head to Kynesgrove, the location of an ancient dragon burial, where you will prove to her you can slay a dragon.

Tell her you can't go yet, and she says she will meet you at Kynesgrove. Fast Travel to High Hrothgar to complete the current quest first. Talk to Arngeir. It is now time to be recognized formally as Dragonborn. Follow Wulfgar who will teach you the final word of Unrelenting Force. Your objective now, is to receive the Greybears; greeting. Receive their greetings, and High Hrothgar is now open to you.

This completes the quest "The Horn of Jurgen Windcaller" and starts the next quest, "A Blade in the Dark"