See also Armor Items.
Armor is a character attribute that reduces the damage taken from physical attacks. Most of the time, armor is increased through the use of equipment, but there are a few other effects that can have an impact on armor or simply on a character's survival.
Loosely speaking, armor is what keeps tanks from dying when they get hit. Armor, obviously, is not the only thing keeping tanks alive, but it is certainly an important factor. According to WoWwiki, the armor reduction percentage formula that appears on your character window armor tooltip is:
Armor / (Armor + 400 + 85 * MobLevel) for levels 1-59
Armor / (Armor - 22167.5 + 467.5 * MobLevel) for levels 60+
Example: If a level 70 character were fighting a level 73 raid boss, and they had 12000 armor, then the damage they take would be reduced by 12000 / (12000 - 22167.5 + 467.5 * 73) = 12000 / 23960 = approximately 50%.
If that character were to have 24000 armor, then the damage would be reduced by 24000 / (24000 - 22167.5 + 467.5 * 73) = 24000 / 35960 = approximately 67%.
There are two important things to understand about this formula.
1) If a character adds some amount of armor, and it prevents X% of the damage-before-armor, adding that amount again will not prevent an additional X%. The amount will be smaller than this. (In the example above, a tank with 12000 armor takes half of the original damage, however, adding another 12000 armor does not eliminate the other half.)
2) Although the damage reduction does not scale with each hit, it allows the character to take more hits, which are also reduced in damage. The formula works out so the damage reduction is an amount that increases the character longevity equally at all levels. Let's imagine three characters, a mage, a shaman, and a warrior all have 10,000 health and nobody is available to heal them.
So even though the damage-per-hit does not seem to scale correctly, the character will take the correct amount of additional punishment before falling over.
The effect of armor has an upper bound of preventing 75% of damage, meaning the most armor can do is make a target live four times as long. Against a level 73 raid boss, the most armor a character can use is 35,880.
Armor values have increasing returns with the more you obtain, meaning 10001 armor is better than having 10000. Each point of armor will give you less physical mitigation compared to the last, but it will increase the amount of damage you can absorb before you die. Armor is a stat no tank should overlook and is especially important for druid tanks.
It should go without saying that gear can improve upon a character's armor rating. In particular, shields tend to have a tremendous amount of armor compared to other items. It is not unheard of for a shield to have more armor by itself than a leather-wearer altogether.
Characters can also gain armor from a variety of effects such as Sanctity Aura and various potions. Each point of agility a character adds will increase their armor by 2 points. The most important of these is bear form, which increases all armor the druid has by 180%, and later 400% for Dire Bear Form. (level 40)
Armor is not the only way to keep a character alive longer until heals can arrive. Some important other tools tanks use include: