Normalization is the adjustment of a formula mechanic from a variable to a constant, in order to regulate the effect from being wildly uncontrollable to being very normalized.
Normalization is a word many players hate, since it has often been associated with nerfing of abilities that were too strong. Before Burning Crusade, many warriors were using a very slow weapon called the Arcanite Reaper. They weren't using it because it was designed to be the best weapon; they were using it because it was SLOW. That meant the damage they dealt with it was very high due to attack power contributions. In theory, the speed of a weapon shouldn't affect the damage that is done with normal attacks, however the Arms ability Mortal Strike was an instant attack that could be spammed to produce damage very quickly...and it did damage equal to the damage that a normal attack would deal.
In other words, a warrior with a 2.0 speed weapon that has 60 DPS, and with 700 attack power, would deal roughly 2.0 * (60 + 700/14) = 220 damage per attack.
However, that same warrior with a similar 60 DPS weapon that had an attack speed of 4.0 would deal 4.0 * (60 + 700/14) = 440 damage per attack.
This, by itself, causes no problem:
Time 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Fast X X X X X X X X X X X Slow X X X X X X
Even though the slow weapon hits twice as hard, it also does it half as often. Overall, the two weapons should do the same damage in the long run.
The issue was with Mortal Strike. Mortal Strike at level 60 deals "weapon damage plus 160", so the fast weapon would deal 380 with Mortal Strike, while the slow weapon would deal 600. If a warrior had enough rage saved up to use the ability 4 times in a row, they would be able to use up that rage in 4.5 seconds. In that time frame, the fast weapon would deal 380 * 4 = 1520 damage, and the slow weapon would do 600 * 4 = 2400. With all the additional bonuses for crits and other talent effects, it was not unheard of for a warrior to one-shot other players. Not only was this unfair, but Blizzard also wanted to encourage players to upgrade their weapons, rather than using a crafted blue weapon just because it happened to be very slow.
Their solution was to "normalize" Mortal Strike. Even though the formula for finding damage is unchanged -- WeaponSpeed * (DPS + AttackPower/14) -- Mortal Strike uses a fixed number for the Weapon Speed instead of whatever the speed of the weapon actually is. This means that the fast weapon and the slow weapon will deal the same damage, and speed shouldn't make a difference.
Mortal Strike, Sinister Strike, and later Aimed Shot, as well as nearly all other Warrior, Rogue, and Hunter instant abilities were normalized. As a result of these normalizations, they use the following weapon speeds for determining their damage:
It should be noted that there are a few abilities that are not normalized that meet the same, such as Stormstrike and Slam. The complete list of abilities for what is and is not normalized is listed at WoWwiki.
Most abilities added after Burning Crusade had their formulas written in such a way as to not need to worry about this contribution. For example, Steady Shot originally did damge equal to "unmodified weapon damage" plus 20% of the hunter's ranged attack power. This is effectively the same as using the 2.8 speed listed for ranged weapons.
While "normalize" usually has negative connotations, it has been used to simply control or regulate the way the game plays. Other factors in the game have been normalized, either directly or indirectly. Many of these issues continue to be related to attack speed.
For example, except for Rifle of the Stoic Guardian, no ranged weapons have been added to the game since the initial launch of Burning Crusade that did not have an attack speed between 2.6 and 3.1 seconds. In the past, their speeds had ranged from 1.7 to 3.4, which created strange variations in hunter damage outputs due to a tremendous number of other factors. Eventually Blizzard decided to just regulate the effects so a player would not have to be a theorycraft master to do a reasonable amount of damage.
Hunter pets mostly attacked and moved at the same rate, however there were a few special examples. Because of these, players were only using one of a very few select pets. In order to make a greater variety of pets viable, special pets with advantages unavaiable to other pets, like Broken Tooth and Spire Spiderling were adjusted so all pets would be (relatively) equal.