You will usually get the  Daedric quest "Discerning the Transmundane" from  Urag gro-Shub in the College of Winterhold while searching for the Elder Scroll in the quest "Elder Knowledge". But you can also get this quest at any other time before doing Elder Knowledge. If you wish to do this quest before Elder Knowledge, skip to the part after "Your objective is now to find Septimus Signus"

Dawnguard Update: If you are playing the Skyrim Dawnguard DLC, you can also get the this quest while doing "Scroll Scouting"
After reading the book "Ruminations on the Elder Scrolls", it appears it was written by a madman.
I've been given a book of Bizarre ramblings about Elder Scrolls, written by Septimus Signus. He claims to have spent time at the Winterhold Arcanaeum.

Objectives: Ask Urag about the insane book

Tell Urag that the "Ruminations" book is incomprehensible. He will tell you it's the work of Septimus Signus. Ask him if he is dead. He will tell you that he hopes not. Was obsessed with Dremer. Took out north and claimed to have found some artifacts. Somewhere in the Ice Fields if you want to try and find him.

"I've learned that Septimus Signus, a preeminent scholar of the Elder Scrolls, is currently living in the far north, studying a dwarven artifact. He might know where I can find an Elder Scroll for Paarthurnax."

Your objective now is to find Septimus Signus.

Go to the town of Winterhold. From the town center, head north to the direction of the college. But don't go up the ramp leading to the college, instead take the road to the right, out of town from its NE side. Head down the mountain, then cross the river towards the north, by either swimming or hopping on the ice. You will arrive at Septimus Signus's Outpost. Enter through the wooden door.

Locate Septimus inside his outpost and talk to him. He does sound crazy. Tell him you heard he knows about Elder Scrolls. He will tell you he knows of one, forgotten. Ask him where it is. He will mumble some nonsense. Ask him if he can help you get the Elder Scroll or  not. He wants something in return. He also needs information from the Elder Scroll to open a Dwemer Lockbox. He wants you to go to Blackreach. Ask him where it is, he will mention the Tower Mzark. He will mention a hidden key. Ask him how to get in. He will give you a sphere and a cube - an attunement sphere and a blank lexicon.

Your quest log should be updated:
"I met Septimus Signus, a brilliant but mad scholar living in the northern ice fields. He's directed me to a dwarven observatory that supposedly houses an Elder Scroll. He doesn't care about the scroll itself, but wants me to use some kind of machine there to inscribe a dwarven lexicon with the scroll's knowledge and bring it to him"

Your goal now is to Transcribe the Lexicon. The Objective on the Elder Knowledge quest, if you are on it, should also update to recovering the Elder Scroll.

Exit the outpost and fast travel to Winterhold, then travel SW to Alftand, using the path behind Jarl's Longhouse. If you have been to Saarthal you can fast travel there to save you some time. Continue SW until you reach Alftand. Then head down the wooden planket bridges and into the Alftand Glacial Ruins.

Go down the tunnels, towards the southeast. Turn to the NE at the room with the pipe. Reach a chamber with a locked barred gate to the NE. Read the research notes on the table. Go south, then West. Watch out for the enemy spiders. Turn south, then go down west again. You will run into a Khajiit named J'darr who will attack you. Kill him. Read Jdarr's Journal. Continue past a room in which some dwarven spheres will attack you. Reach a room with a novice lock gate protecting some treasure, get it if you want.

Open a golden door to the NW. At the junction with the chest, go South. In a room to the west, an apprentice golden door lock hides a dwarven sphere protecting some treasure. Pick the lock if you want. Then keep heading south.

Turn east, then navigate the hall with the pistons counterclockwise, avoiding the pistons as they push out. Head east and enter the Alftand Animonculory. Keep going east, past a gold door. Climb the stairs. Watch for the blade trap on the ramp ahead. Pull the lever to open the gate to the north. Work your way down the spiraling ramp. When it comes to an end in front of some gold novice door, jump down to the pipe then jump down again to another spiral ramp, battling falmers along the way.

Go past the flames trap to the NE and open the door. Climb down the stairs and defeat the two Falmers in the room with the spinning cogs. Exit that room to the west, reach another room with some falmer, fire traps, and steep stairs. Go down the stairs and then go down some more stairs to the south. Enter the elevator and pull the switch to go up to the Alftand Glacial Ruins. Activate the switch to open the barred gate on the south to the room in which you found the research notes. This will now let you use the elevator to travel easier from the Glacial Ruins to Alftand Animonculory when you return to the dungeon in the future. Take the elevator back down.

Go SE and  head down the spiral platform. Exit through a corridor to the NE. Open the door to the Alftand Cathedral. Go west to enter a grotto, climb the stairs to a platform with some chests and a lever. Activate the lever. Go through the gate that opened, defeat the giant Dwarven Centurion. Loot the key to Alftand Lift from him.

Climb the stairs and open the gate to the SW. Kill the two thieves,  Sulla and Umana. There is a Master locked gate to the SW, but your Alftand key will open it. Take the lift up to Skyrim and activate the lever to unlock this elevator for future easy access to the Alftand Cathedral. Take the elevator back down to the Catehdral. Activate the Dwarven Mechanism using the Attunement Sphere from Septimus. This reveals a secret stairway below you. Go down and open door to Blackreach.

This is a huge place, with many activities, but this walkthrough will focus on getting the Elder Scroll. Although first, head over to the SE end of the cave and locate a gold head statue with spinning cogs on the side and a button on top. Activate the button. This will open an elevator to Skyrim. Enter and take the lift up. Activate the lever, and you will now have access to  the Great Lift at Raldbthar west of Windhelm. Take the elevator back down to Blackreach.

Go west past a lake and a waterfall. Reach another lake surrounding a tall structure with stairs. Jump into the water and swim around to access the stairs, climb them to the top and open the door. Pull the lever to take the lift up to the Tower of Mzark. Open the door to the SW and enter the Oculory. This is a PUZZLE

Solution to Tower of Mzark Puzzle:
Climb up to the top of the platform and activate the Lexicon Receptable to the right of the buttons using the Blank Lexicon from Septimus. There are 5 pedestals with buttons on them. We will label them, from left to right, A, B, C, D and E.

Pedestals C, D and E are currenly lit. Press Button D four times. This will activate Button B. The blank lexicon will also start to glow and rotate.

Press Button B now twice. This activates Button A. Press Button A.

So the complete sequence is: D,D,D,D,B,B,A
(Notice that each button was pressed the number of times corresponding to the pedestal position, which is the solution to this riddle)

This will lower a crystal onto the platform in the center of the room. The Blank Lexicon has also settled and is now blue and has been transcribed. Pick up the Runed Lexicon.

Now go to the center of the chamber and take the Elder Scroll.

Take the door to the east, and use lever to ride the elevator up to Skyrim.

This will complete the quest "Elder Knowledge" if you are on it, and commence "Alduin's Bane". Most of the people reading this walkthrough are in the Tower of Mzark as part of the Main Quest, Elder Knowledge. At this time, you want to return to  Paarthurnmax on the top of the mountain at the Throat of the World.

Refer to the second part of this walkthrough for instructions on how to complete the second half of "Discerning the Transmundane" after proceeding with the main quest.