Oblivion:Hermaeus Mora

Quick Walkthrough
# Once you've met the requirements, speak with Casta Flavus, or activate the shrine directly.
# Hermaeus Mora will instruct you to collect one soul of each race.
# You will receive ''Hermaeus Mora's Soul Gem and the Mora's Soul Trap spell to complete this task.
# Once all souls have been collected, return to the shrine and activate it.
# You are rewarded with the Oghma Infinium'' which will permanently boost a series of stats that you must choose from a list.

Detailed Walkthrough
Locating the Shrine of Hermaeus Mora
If the requirements have been met, you can proceed directly to the shrine and begin.  If you have not yet met the requirements, you will be told by Ri'Jirr: "You have not yet followed the path required to speak with the Lord Hermaeus Mora".  Unlike all other shrines, you will not be given any clues as to what is required, nor will a quest be added to your journal.  Once the requirements have been met, Casta Flavus will appear the next time you sleep and inform you that you are now ready to start Hermaeus' quest.  

Before you can activate the shrine and start this quest, you must meet the following requirements:
You must have completed all 14 of the other Daedric Shrine quests.
You must have at least started the Blood of the Daedra quest (the point in the main quest where Martin asks you for a daedric artifact).  It is possible to have finished the quest, but that is not a requirement (in other words, it is possible to use this quest's reward, the Oghma Infinium, as Martin's daedric artifact).
You can do this quest even if you keep Umbra in Clavicus Vile's quest (although you must visit Clavicus Vile's shrine and tell him that you will not give him Umbra, thus completing the quest).

The Shrine itself is located on a remote plateau deep within the northwestern portion of the Jerall Mountains.  It is due west of Cloud Ruler Temple and due north of Sancre Tor.  Northwest of Sancre Tor, at the top of the long flight of stone steps, there is an unmarked road that leads straight to the shrine.  The road's stairs and bridges make it possible to easily traverse the mountains, but several enemies will be found along the road.  No NPC will give directions to it.

Once you have found the shrine, activate the statue.  Hermaeus Mora's statue is probably by far the most disturbing and mystifying of all the Daedric princes. He is depicted as a rotund bulbous mass; with four scorpion-like claws, as well as several tentacles and eyeballs. He speaks in an alarming, almost guttural voice:
:I see you have served my brethren, and now you come to serve me. I am Hermaeus Mora. Past, present, and future are as one to me. My followers seek to perform a divination. For this, they need souls, one of each of the sentient races. You will provide these. I give you a spell to trap these souls. Return to me when you have finished. When you do, I will further your journey on your true path.

The Task at Hand
Hermaeus Mora will ask you to capture one soul for each of the 10 races of Tamriel.  He will give you ''Hermaeus Mora's Soul Gem and the Mora's Soul Trap'' spell.  The soul gem will only work for this quest and it can only capture souls that are trapped with this unique spell.  This spell is cheap to cast and lasts 120 seconds.

One way to complete this quest if you don't want to murder innocent people is to go vampire hunting: Vampires are not undead, and are classified as NPCs, therefore their souls count.  However, Argonian and Khajiiti vampires do not exist. Try the Vampire Dungeon page for a list of places where you can steal vampire souls.

If you haven't started fighting in the Arena yet, this would also be a good opportunity to gain all of the souls. Until you get promoted to Warrior, which is the fourth rank, every race appears at least once.

You can also use the Skull of Corruption on any non-essential NPC and kill the clone before it disappears. This will, however, incur a bounty because using the Skull counts as assault.

Some places where you can find NPCs of multiple races include:
Daedric Shrines: Representatives of each race can be found at the various Daedric shrines, and they are generally easy kills, since most have no armor and/or weak weapons.  Other than Olyn Seran at Molag Bal's shrine (who is the master trainer of Conjuration), the Daedra Worshippers no longer serve any purpose in the game.
Rockmilk Cave between Bravil and Leyawiin.  The large number and different types of bandits and marauders fighting likely results in all races, except for Bretons, being present.

Each of the following links leads to a page listing all named NPCs of that race in the game; also provided are generic NPC enemies that can be of each race.
Altmer: Necromancers, conjurers, and marauder battlemages.
Argonian: Marauder archers.  Also, Veyond Cave (north of Bravil) is filled with hostile Argonian tribesmen.
Bosmer: Bandit bowmen.
Breton: Necromancers, conjurers, and bandit hedge wizards.
Dunmer: Bandit bowmen and ringleaders.
Imperial: Marauder archers, warlords, and the Hackdirt Brethren.
Khajiit: Bandits and highwaymen.
Nord: Marauders.
Orc: Marauders.
Redguard: Bandits.

Return to the Shrine
Once you gather all of the souls and activate the shrine again, Hermaeus Mora gives you a special book, the Oghma Infinium, as a reward and says:
:You have served me well, and I am pleased. I give you this, the Oghma Infinium. Knowledge given by me to Xarxes is recorded within. Read it, and gain knowledge of yourself. Go now. Learn, and remember.
When read, the Oghma Infinium presents a choice of three paths: spirit, shadow, or steel.  You may choose only one path.  Each path permanently boosts two attributes and three skills by 10 points each.  One path is bugged, see this page for more information.

Notes
It is possible to trigger this quest without receiving Daedric Artifacts from each of the other 14 Daedric quests; officially failing a Daedric quest also counts.  For example, if Melus Petilius dies, Molag Bal's quest immediately ends without any chance of a reward; however, Molag Bal's quest still counts as far as Hermaeus Mora is concerned.  In general, as long as the Daedric quest is in your list of completed quests, it will count.  The two exceptions are Malacath's and Namira's quests, which must be finished correctly (see those quests for details).
Your Quest log will update for each soul you capture; if it doesn't update when you are sure you had soul trapped a new race then your game has become bugged.
The Oghma Infinium is a Quest Item and cannot be removed from your inventory until you use it.
''Mora's Soul Trap is not a true Soul Trap spell: it does not trap creatures in standard soul gems and it cannot be used to create Black Soul Gems.  Furthermore, even though standard Soul Trap spells are in the school of Mysticism, Mora's Soul Trap is in the school of Alteration, and will give you Alteration skill points when used.

Bugs
 

There are a couple of bugs associated with this quest's reward. These bugs are detailed on the article about the Oghma Infinium.

Journal Entries