An Elder Scroll will hold the knowledge Paarthurnax requires in order to send you back in time to learn the Dragonrend shout from the ancient Nords who used it to vanquish Alduin thousands of years ago. He doesn’t know where to find one, but suggests some people who might.
Go and ask either Arngeir the Greybeard monk or Esbern the Blade archivist where you might find an Elder Scroll. Both will point you in the same direction: to the College of Winterhold. Travel to the city of Winterhold when you are ready to continue with this quest.
When you get to the entrance to the college, the mage standing guard at the entrance will ask you to demonstrate your magical power by casting a magic spell. You can oblige her if you want, or you can tell her you’re a dragonborn and demonstrate one of your shouts, which will result in her granting you admittance to the college.
Once inside the college, speak with the librarian, an Orc named Urag gro-Shub, who tells you of a man who knows much about the Elder Scrolls. Travel north to find Septimus Signus at his outpost, using your map to guide you. He will tell that an Elder Scroll can be found within an ancient, Dwemer ruin, and give you two items which will function as keys later on.
The Dwemer ruin Septimus spoke of is known as Alftand. It is a very large complex that is also the gateway to Blackreach, a tremendous underground cavern. There are many side objectives within Blackreach, so this quest can be quite a lengthy one.
Within Alftand and Blackreach you will face two particularly dangerous enemies: Steam Centurions and Falmer. Steam Centurions are the most powerful of the Dwemer constructs, and were designed in the form of a large armored warrior, with a hammer on one arm and a spring-loaded spike in the other, and they can vent superheated steam at a distance. Falmer are the near-mythical Snow Elves, once thought to have been hunted to extinction by the ancestors of the Nords. They now live deep underground and have devolved into primitive, disfigured, violent remnants of their former selves. Both of these enemies are capable of making this quest nigh-impossible if adequate preparations are not made.
The easiest way to attack this lengthy and troublesome mission is to prepare yourself for a multitude of difficult battles. So make sure you’re well stocked on food, potions, scrolls, and anything else you think you’ll need. Also buy any spells that look useful from the wizards in the college before you depart (the Clairvoyance spell can come in handy when navigating Blackreach). Definitely take a follower with you.
First, you must make your way through Alftand. It is a large dungeon with a lot of fighting, but it will always be evident where you have to go. Eventually you will come to a door, which you will walk out of to encounter a small group of Falmer. Kill the Falmer, and on each side of you will be two staircases. Walk up the staircase, and you will find a lever which controls the gate below.
On the other side of the gate you will find a Dwarven Centurion, which can prove to be a challenge to low/mid level characters. After you destroy it, proceed up some more stairs where you will encounter two people who will attack you. It is possible to have these two people attack each other AND the centurion. They will finish of the centurion together, then fight each other to the death, leaving you can kill whichever one is left. You can alternatively use the sphere given to you by Septimus Signus on a mysterious table to reveal another staircase which allows you to avoid the two people.
Finally, you will arrive in the vast underground cavern of Blackreach. This is a very difficult place to get through if you’re at a low level. There are many Falmer and Dwemer robotics. A quest marker will be present on your map, and it will look like it’s telling you to visit a mountain range, but Blackreach is VERY LARGE. The marker is IN Blackreach. It will take a while for you to trudge your way through. Follow the cobblestone path to the right. If you’re looking to avoid confrontations with enemies, avoid the buildings, as they seem to contain the greatest concentrations of enemies here.
When you reach the marker, you will find the Tower of Mzark, which seems to be some sort of an observatory. Look around the big Dwemer containment structure to find a little podium with buttons on it. The order in which the buttons need to be pressed seems to be different for everyone, but it will often work to do this: press the third button (assume they are numbered 1, 2, 3 and 4 from left to right) until the second button lights up. Then continue to press the second button until the first lights up. Press the first, and the Elder Scroll will be revealed (if that process doesn’t work, just play with the buttons until you get it right. It sometimes takes only five minutes to figure out the combination). Pick up the transcribed Lexicon and the Elder Scroll. Now use the elevator-like structure to get out of Blackreach.
WARNING: Attempting to read the Elder Scroll will inflict temporary blindless upon your character.
Take the Elder Scroll to Paarthurnax atop the Throat of the World when you’re ready to proceed with the quest.